Scheme

📁 Source: Mathlib/AlgebraicGeometry/Scheme.lean

Statistics

Metric	Count
Definitions Scheme, toLRSHom, app, appLE, appTop, arrowStalkMapIsoOfEq, copyBase, homeomorph, stalkMap, toLRSHom, toLRSHom', SpecΓIdentity, algebra_section_section_basicOpen, arrowStalkMapIsoOfEq, basicOpen, delabAdjoinNotation, delabCoeFunAppNotation, delabCoeFunNotation, empty, forget, forgetToLocallyRingedSpace, forgetToTop, forgetToTop_obj_eq_coe, forget_obj_eq_coe, fullyFaithfulForgetToLocallyRingedSpace, hasCoeToTopCat, homeoOfIso, instCategory, instCoeFunHomForallCarrierCarrierCommRingCat, instCoeSortType, instEmptyCollection, instPreorderCarrierCarrierCommRingCat, instUniqueCarrierCarrierCommRingCatSpecOf, sheaf, toLocallyRingedSpace, zeroLocus, Γ, ΓSpecIso, map, «term_⁻¹ᵁ_», «termΓ(_,_)»»), schemeIsoToHomeo, «termSpec(_)»»)	43
Theorems appLE_comp_appLE, appLE_comp_appLE_assoc, appLE_congr, appLE_eq_app, appLE_map, appLE_map', appLE_map'_assoc, appLE_map_assoc, app_eq, app_eq_appLE, coe_preimage, comp_app, comp_appLE, comp_appLE_assoc, comp_appTop, comp_appTop_assoc, comp_app_assoc, comp_apply, comp_base, comp_base_assoc, comp_preimage, comp_toLRSHom, comp_toLRSHom_assoc, congr_app, continuous, copyBase_eq, eqToHom_app, ext, ext', germ_stalkMap, germ_stalkMap_apply, germ_stalkMap_assoc, homeomorph_apply, iSup_preimage_eq_top, id_app, id_appTop, id_base, id_preimage, instIsIsoCommRingCatApp, instIsLocalHomCarrierStalkCommRingCatPresheafCoeContinuousMapCarrierCarrierHomTopCatBaseRingHomHomStalkMap, inv_app, inv_appTop, isIso_base, isIso_toLRSHom, isIso_toPshHom, map_appLE, map_appLE', map_appLE'_assoc, map_appLE_assoc, mem_preimage, naturality, naturality_assoc, preimage_basicOpen, preimage_basicOpen_top, preimage_bot, preimage_iSup, preimage_iSup_eq_top, preimage_inf, preimage_le_preimage_of_le, preimage_mono, preimage_sup, preimage_top, stalkMap_comp, stalkMap_congr, stalkMap_congr_assoc, stalkMap_congr_hom, stalkMap_congr_hom_assoc, stalkMap_congr_point, stalkMap_congr_point_assoc, stalkMap_hom_inv, stalkMap_hom_inv_apply, stalkMap_hom_inv_assoc, stalkMap_id, stalkMap_inv_hom, stalkMap_inv_hom_apply, stalkMap_inv_hom_assoc, stalkSpecializes_stalkMap, stalkSpecializes_stalkMap_apply, stalkSpecializes_stalkMap_assoc, SpecMap_presheaf_map_eqToHom, Spec_map, Spec_map_presheaf_map_eqToHom, Spec_obj, SpecΓIdentity_app, SpecΓIdentity_hom_app, SpecΓIdentity_inv_app, appLE_comp_appLE, app_eq, basicOpen_add_le, basicOpen_appLE, basicOpen_le, basicOpen_mul, basicOpen_of_isUnit, basicOpen_one, basicOpen_pow, basicOpen_res, basicOpen_res_eq, basicOpen_restrict, basicOpen_zero, codisjoint_zeroLocus, coe_homeoOfIso, coe_homeoOfIso_symm, comp_app, comp_appLE, comp_appLE_assoc, comp_appTop, comp_appTop_assoc, comp_app_assoc, comp_base, comp_base_apply, comp_base_assoc, comp_coeBase, comp_coeBase_assoc, comp_toLRSHom, comp_toLRSHom_assoc, congr_app, default_asIdeal, empty_carrier_carrier, empty_presheaf, eqToHom_c_app, forgetToLocallyRingedSpace_map, forgetToLocallyRingedSpace_obj, forgetToTop_comp_forget, forgetToTop_map, forgetToTop_obj, forget_map, forget_obj, fullyFaithfulForgetToLocallyRingedSpace_preimage_toLRSHom, height_of_isClosed, hom_base_inv_base, hom_base_inv_base_assoc, hom_inv_apply, homeoOfIso_apply, homeoOfIso_symm, base, id_app, id_appTop, instFaithfulLocallyRingedSpaceForgetToLocallyRingedSpace, instFullLocallyRingedSpaceForgetToLocallyRingedSpace, instSubsingletonCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensBot, inv_app, inv_appTop, inv_base_hom_base, inv_base_hom_base_assoc, inv_hom_apply, isEmpty_of_commSq, iso_hom_base_inv_base, iso_hom_base_inv_base_apply, iso_inv_base_hom_base, iso_inv_base_hom_base_apply, le_iff_specializes, local_affine, mem_basicOpen, mem_basicOpen', mem_basicOpen'', mem_basicOpen_top, mem_zeroLocus_iff, preimage_basicOpen, preimage_basicOpen_top, preimage_comp, preimage_zeroLocus, presheaf_map_eqToHom_op, stalkMap_comp, stalkMap_congr, stalkMap_congr_assoc, stalkMap_congr_hom, stalkMap_congr_hom_assoc, stalkMap_congr_point, stalkMap_congr_point_assoc, stalkMap_germ, stalkMap_germ_apply, stalkMap_germ_assoc, stalkMap_hom_inv, stalkMap_hom_inv_apply, stalkMap_hom_inv_assoc, stalkMap_id, stalkMap_inv_hom, stalkMap_inv_hom_apply, stalkMap_inv_hom_assoc, stalkSpecializes_stalkMap, stalkSpecializes_stalkMap_apply, stalkSpecializes_stalkMap_assoc, toOpen_eq, zeroLocus_def, zeroLocus_empty_eq_univ, zeroLocus_iUnion, zeroLocus_isClosed, zeroLocus_map, zeroLocus_map_of_eq, zeroLocus_mono, zeroLocus_mul, zeroLocus_radical, zeroLocus_setMul, zeroLocus_singleton, zeroLocus_span, zeroLocus_univ, ΓSpecIso_inv, ΓSpecIso_inv_naturality, ΓSpecIso_inv_naturality_assoc, ΓSpecIso_naturality, ΓSpecIso_naturality_assoc, Γ_def, Γ_map, Γ_map_op, Γ_obj, Γ_obj_op, map_app, map_appLE, map_apply, map_base, map_base_apply, map_comp, map_comp_assoc, map_eqToHom, map_id, map_inv, SpecMap_preimage_basicOpen, Spec_carrier, Spec_closedPoint, Spec_presheaf, Spec_sheaf, Spec_toLocallyRingedSpace, Spec_zeroLocus, Spec_zeroLocus_eq_zeroLocus, basicOpen_eq_of_affine, basicOpen_eq_of_affine', germ_eq_zero_of_pow_mul_eq_zero, instIsIsoSchemeMapOfCommRingCat, instNonemptyCarrierCarrierCommRingCatSpecOfNontrivialCarrier	229
Total	272

AlgebraicGeometry

Definitions

Name	Category	Theorems
Scheme 📖	CompData	1660 math math: Scheme.Pullback.gluedLiftPullbackMap_snd, continuousMapPresheaf_obj, AffineSpace.not_isIntegralHom, Scheme.Hom.resLE_map_assoc, Surjective.instFstScheme, AffineSpace.map_Spec_map, Scheme.mem_propQCPrecoverage_iff_exists_quasiCompactCover, Scheme.GlueData.oneHypercover_I₁, instIsStableUnderBaseChangeSchemeGeometricallyIrreducible, WeaklyEtale.weaklyEtale_eq_flat_inf_diagonal_flat, Scheme.coprodPresheafObjIso_hom_fst_assoc, Scheme.Hom.toPartialMap_hom, descendsAlong_isOpenImmersion_surjective_inf_flat_inf_quasicompact', Scheme.instEffectiveEpiOfLocallyOfFinitePresentationOfSurjectiveOfFlat, Scheme.map_PrimeSpectrum_basicOpen_of_affine, Scheme.bot_mem_grothendieckTopology, Scheme.Modules.pseudofunctor_map_adj, Scheme.coverOfIsIso_I₀, IsImmersion.isStableUnderBaseChange, instLocallyQuasiFiniteSndScheme, Scheme.isoOfEq_inv, Scheme.Cover.pullbackCoverOverProp_I₀, IsAffineOpen.isoSpec_inv_appTop, Spec.map_eq_id, AffineSpace.map_toSpecMvPoly_assoc, Scheme.AffineOpenCover.openCover_f, Scheme.Cover.RelativeGluingData.cover_X, Scheme.Modules.pushforwardId_inv_app_app, IsClosedImmersion.instIsIsoSchemeToImage, AffineSpace.map_reindex, IsAffineOpen.map_fromSpec_assoc, Scheme.Pullback.Triplet.specTensorTo_fst, instIsAffineXSchemeCover, Scheme.Hom.opensRange_pullbackSnd, Scheme.Cover.pullbackCoverOver_X, IsProper.instCompScheme, Scheme.Cover.gluedCoverT'_snd_snd_assoc, instCompactSpaceCarrierCarrierCommRingCatEqualizerSchemeOfQuasiSeparatedSpace, Scheme.Spec_map_stalkMap_fromSpecStalk, Scheme.Cover.LocallyDirected.directed, Scheme.ProEt.instIsContinuousCompOverForgetForgetTopologyProetaleTopology, descendsAlong_universallyClosed_surjective_inf_flat_inf_quasicompact, Scheme.Pullback.gluedLiftPullbackMap_snd_assoc, instIsAffineXSchemeFiniteSubcover, Scheme.PartialMap.ext_iff, instHasOfPostcompPropertySchemeUniversallyClosedIsSeparated, pointOfClosedPoint_comp_assoc, Scheme.isEmpty_pullback_iff, instIsReducedPullbackSchemeOfGeometricallyReducedOfFlatOfIsLocallyNoetherian, AffineSpace.isoOfIsAffine_inv, IsSeparated.isStableUnderBaseChange, Scheme.Hom.quasiFiniteLocus_comp, isCommMonObj_of_isProper_of_geometricallyIntegral, instIsAffinePullbackSchemeOfIsAffineHom_1, Scheme.IdealSheafData.glueDataObjHom_ι_assoc, instIsLocallyArtinianXScheme, QuasiCompactCover.of_finite, opensCone_pt, Scheme.Cover.LocallyDirected.ofIsBasisOpensRange_le_iff, Proj.basicOpenIsoSpec_inv_ι_assoc, coprodSpec_apply, RingHom.IsStableUnderBaseChange.pullback_fst_appTop, instCompactSpaceCarrierCarrierCommRingCatPullbackSchemeOfQuasiCompact, Scheme.Hom.toNormalization_inl_normalizationCoprodIso_hom_assoc, Scheme.AffineCover.cover_I₀, Scheme.Hom.toNormalization_normalizationPullback_fst, instLocallyQuasiFiniteCompSchemeιQuasiFiniteLocus, IsOpenImmersion.hasLimit_cospan_forget_of_right', instIsIsoSchemeCoprodComparisonOppositeCommRingCatSpec, Etale.instFstScheme, IsAffineOpen.map_fromSpec, ΓSpec.isIso_adjunction_counit, Scheme.germ_stalkClosedPointTo_Spec_fromSpecStalk_assoc, instQuasiCompactιSchemeOfQuasiSeparatedSpaceCarrierCarrierCommRingCat, injective_isStableUnderComposition, Scheme.affineBasisCover_is_basis, instIsZariskiLocalAtSourceDiagonalSchemeOfHasOfPostcompPropertyOfRespectsRightIsOpenImmersion, Scheme.Opens.toSpecΓ_naturality, instLocallyQuasiFiniteFstScheme, Scheme.Opens.toSpecΓ_SpecMap_presheaf_map_top, Scheme.Pullback.openCoverOfLeftRight_f, Scheme.comp_toLRSHom_assoc, Scheme.restrictFunctorΓ_inv_app, Scheme.Hom.inr_toNormalization_normalizationCoprodIso_inv, Scheme.mem_grothendieckTopology_iff, preservesColimit_Γ, instPreservesLimitSchemeOppositeCommRingCatRightOpΓOfIsAffineHomMapOfCompactSpaceOfQuasiSeparatedSpaceCarrierCarrierObj, Scheme.Pullback.Triplet.ofPoint_x, Scheme.OpenCover.map_glueMorphismsOverOfLocallyDirected_left_assoc, Scheme.instQuasiCompactCover, Scheme.overGrothendieckTopology_eq_toGrothendieck_overPretopology, universallyClosedTypeComp, quasiSeparated_comp, HasAffineProperty.affineAnd_le_affineAnd, Proj.fromOfGlobalSections_toSpecZero_assoc, Scheme.Hom.exists_isIso_morphismRestrict_toNormalization, AffineSpace.toSpecMvPolyIntEquiv_symm_apply, morphismRestrict_ι_assoc, Scheme.fromSpecStalk_toSpecΓ_assoc, Scheme.Modules.pushforwardCongr_hom_app_app, AffineSpace.SpecIso_hom_appTop, sourceLocalClosure.instIsMultiplicativeSchemeOfIsStableUnderBaseChange, exists_isAffineOpen_preimage_eq, Scheme.instQuasiCompactCoverQculift, IsSeparated.isClosedImmersion_diagonal, instIsAffineSigmaObjScheme, Scheme.zariskiTopology_le_etaleTopology, isIso_SpecMap_iff, Scheme.zariskiTopology_eq, Scheme.Cover.copy_f, Scheme.Cover.iUnion_range, IsDominant.respectsIso, QuasiCompactCover.instCoverOfIsIso, pullbackRestrictIsoRestrict_hom_morphismRestrict_assoc, Spec.map_surjective, IsProper.instFstScheme, weaklyEtale_iff, isCompact_iff_exists, IsAffineOpen.isCompact_pullback_inf, instFinitaryExtensiveScheme, pointsPi_surjective_of_isAffine, Scheme.Pullback.t'_snd_fst_fst_assoc, isOpenImmersion_isStableUnderComposition, Scheme.Cover.glued_cover_cocycle_fst, Scheme.instIsStableUnderCompositionQcPrecoverage, IsSeparated.instIsClosedImmersionMapDescScheme, sigmaOpenCover_X, GeometricallyIrreducible.comp, Scheme.Hom.iUnion_support_ker_openCover_map_comp, UniversallyOpen.instIsStableUnderBaseChangeScheme, diagonal_SpecMap, AffineTargetMorphismProperty.IsLocal.respectsIso, Scheme.inv_app, Scheme.Pullback.openCoverOfBase_I₀, Scheme.Hom.id_appTop, isAffineHom_diagonal_iff, ι_right_coprodIsoSigma_inv, ι_sigmaSpec, Scheme.proetalePrecoverage_le_precoverage_weaklyEtale, Scheme.emptyTo_base, instHasColimitOverSchemeTopMorphismProperty, Scheme.Cover.instIsIsoFromGlued, Scheme.Cover.copy_X, Scheme.Hom.le_ker_comp, Spec.map_injective, coprodSpec_inr, IsOpenImmersion.comp_lift_assoc, IsProper.instSndScheme, Scheme.OpenCover.finiteSubcover_f, instIsIsoSchemeSigmaSpecOfFinite, Scheme.Hom.singleton_mem_fpqcPrecoverage, Scheme.Spec_map_stalkSpecializes_fromSpecStalk, Scheme.Hom.toNormalization_normalizationDesc, Scheme.Pullback.openCoverOfBase_f, HasRingHomProperty.comp_of_isOpenImmersion, Scheme.ideal_ker_le_ker_ΓSpecIso_inv_comp, Scheme.Cover.pullbackCoverOver'_X, Scheme.stalkMap_congr_assoc, Scheme.Pullback.openCoverOfRight_I₀, Scheme.Opens.toSpecΓ_SpecMap_map, Scheme.isoSpec_inv_toSpecΓ, Scheme.Pullback.SpecTensorTo_SpecOfPoint, UniversallyInjective.iff_diagonal, instIsStableUnderCompositionSchemeSmooth, Scheme.hom_inv_apply, pointsPi_injective, Scheme.Hom.instIsIsoToNormalizationOfIsIntegralHom, Scheme.Cover.ColimitGluingData.pullbackGluedIso_inv_snd_assoc, isNoetherian_iff_of_finite_affine_openCover, Scheme.instHasPullbacksPrecoverageOfHasPullbacks, ExistsHomHomCompEqCompAux.exists_eq, Scheme.ι_toIso_inv, Scheme.locallyCoverDense_of_le, instIsMultiplicativeSchemeQuasiCompact, Scheme.instIsOverFromSpecStalkOfMem, isPullback_opens_inf, Scheme.Cover.functorOfLocallyDirected_obj, Scheme.Cover.ColimitGluingData.isPullback, Scheme.Hom.preimageIso_hom_ι_assoc, Scheme.isoOfEq_inv_ι, isPullback_fiberToSpecResidueField_of_isPullback, Scheme.openCoverOfIsOpenCover_I₀, instIsOpenImmersionLeftSchemeDiscretePUnitMapWalkingSpanOverTopMorphismPropertySpan, Scheme.instQuasiCompactCoverForgetQc, ExistsHomHomCompEqCompAux.h𝒰S, Scheme.Hom.toNormalization_inr_normalizationCoprodIso_hom, UniversallyOpen.instIsMultiplicativeScheme, Scheme.exists_hom_isAffine_of_isLocalAtSource, HasAffineProperty.isStableUnderBaseChange, HasAffineProperty.affineAnd_isStableUnderBaseChange, Scheme.Cover.glued_cover_cocycle, Scheme.Pullback.instHasPullbacks, Scheme.IdealSheafData.glueDataObjHom_ι, Scheme.affineOpenCover_I₀, isImmersion_eq_inf, Scheme.Cover.gluedCover_t, Scheme.Pullback.t_fst_snd_assoc, Scheme.SpecΓIdentity_hom_app, locallyOfFiniteType_isStableUnderBaseChange, Scheme.preimage_comp, instIsZariskiLocalAtTargetMonomorphismsScheme, Scheme.Hom.stalkMap_congr_hom_assoc, Scheme.Cover.RelativeGluingData.toBase_preimage_eq_opensRange_ι, Scheme.quasiCompactCover_shrink_iff, Scheme.Cover.RelativeGluingData.cover_I₀, sourceLocalClosure.iff_forall_exists, Scheme.precoverage_mono, Scheme.Pullback.Triplet.isPullback_SpecMap_tensor, Scheme.forget_map, sigmaMk_mk, Scheme.Pullback.Triplet.specTensorTo_base_snd, Scheme.Hom.isOver_iff, Scheme.PartialMap.fromSpecStalkOfMem_compHom, affineAnd_respectsIso, descendsAlong_isomorphisms_surjective_inf_flat_inf_quasicompact, Scheme.isoSpec_inv_preimage_zeroLocus, Flat.instDescScheme, Scheme.Hom.normalizationDesc_comp_assoc, Scheme.Cover.pullbackCoverOver_I₀, instIsMultiplicativeSchemeIsSchemeTheoreticallyDominant, Scheme.LocalRepresentability.isRepresentable, AffineSpace.map_over_assoc, Scheme.IdealSheafData.inclusion_subschemeι_assoc, Scheme.zariskiTopology_le_fpqcTopology, IsClosedImmersion.comp, Scheme.Modules.pseudofunctor_map_l, Scheme.instIsClosedImmersionιOfIsSeparated, Scheme.Cover.gluedCover_U, Scheme.mem_jointlySurjectiveTopology_iff_jointlySurjectivePretopology, Scheme.exists_hom_comp_eq_comp_of_locallyOfFiniteType, IsSeparated.instIsClosedImmersionLiftSchemeId, Scheme.Pullback.tensorCongr_SpecTensorTo, IsSeparated.eq_valuativeCriterion, Scheme.Hom.SpecMap_residueFieldMap_fromSpecResidueField_assoc, instGeometricallyReducedSndScheme, instCoproductsOfShapeDisjointSchemeOfSmall, Scheme.OpenCover.map_glueMorphismsOfLocallyDirected_assoc, IsImmersion.comp, Scheme.LocalRepresentability.instIsLocallySurjectiveHomYonedaGluedToSheafOfIsLocallySurjectiveZariskiTopologyDescFunctorOppositeType, Scheme.Hom.toNormalization_fromNormalization_assoc, Flat.epi_of_flat_of_surjective, Scheme.forgetToTop_map, UniversallyOpen.out, Scheme.instSubcanonicalFppfTopology, IsClosedImmersion.eq_proper_inf_monomorphisms, Scheme.bot_mem_precoverage, Scheme.Cover.ofQuasiCompactCover_toPreZeroHypercover, IsImmersion.instιScheme, isSmooth_isStableUnderBaseChange, Scheme.coprodPresheafObjIso_hom_snd_assoc, instGeometricallyIrreducibleCompSchemeOfIsOpenImmersionOfSurjective, AffineScheme.mk_obj, Scheme.isoSpec_hom, Scheme.ι_toIso_inv_assoc, Scheme.Hom.id_preimage, Scheme.Modules.pseudofunctor_right_unitality_assoc, Scheme.Hom.toImage_imageι, Scheme.germ_stalkClosedPointTo_Spec_fromSpecStalk, SpecToEquivOfLocalRing_apply_snd_coe, Scheme.OpenCover.pullbackCoverAffineRefinementObjIso_inv_pullbackHom, HasAffineProperty.diagonal_iff, Scheme.forget_obj, Scheme.eqToHom_c_app, Scheme.Pullback.gluedLift_p1, Scheme.exists_affine_mem_range_and_range_subset, instHasOfPostcompPropertySchemeIsClosedImmersionIsSeparated, AffineSpace.homOverEquiv_symm_apply_coe, Scheme.forgetToTop_obj, Scheme.affineOpenCover_idx, Scheme.Opens.isoOfLE_hom_ι_assoc, Scheme.Hom.fromNormalization_preimage, IsIntegralHom.instSndScheme, Scheme.toSpecΓ_naturality_assoc, Scheme.Pullback.Triplet.snd_SpecTensorTo_apply, Scheme.Modules.pushforwardComp_inv_app_app, LocallyQuasiFinite.comp_iff, IsSchemeTheoreticallyDominant.pullbackSnd, IsFinite.comp_iff, HasRingHomProperty.respects_isOpenImmersion, Scheme.Cover.ColimitGluingData.cocone_ι_transitionMap, Scheme.OpenCover.map_glueMorphismsOverOfLocallyDirected_left, Scheme.Pullback.Triplet.specTensorTo_fst_assoc, Scheme.isoOfEq_hom_ι, WeaklyEtale.instFstScheme, IsAffineHom.comp_iff, AffineScheme.forgetToScheme_preservesLimits, HasRingHomProperty.containsIdentities, Scheme.presieve₀_mem_precoverage_iff, Scheme.comp_base, IsClosedImmersion.lift_fac_assoc, AffineSpace.isoOfIsAffine_inv_appTop_coord, Scheme.Pullback.instNonemptyCarrierCarrierCommRingCatPullbackOfSubsingleton, AffineSpace.isoOfIsAffine_inv_over_assoc, isIso_iff_isIso_stalkMap, IsZariskiLocalAtSource.iff_of_iSup_eq_top, Scheme.OpenCover.map_glueMorphismsOfLocallyDirected, Scheme.isoSpec_hom_naturality, instIsEmptyCarrierCarrierCommRingCatEmptyCollectionScheme, Scheme.homOfLE_ι, Scheme.toSpecΓ_isoSpec_inv_assoc, Scheme.Pullback.residueFieldCongr_inv_residueFieldMap_ofPoint, IsOpenImmersion.range_pullbackSnd, Scheme.Hom.resLE_map, Scheme.Cover.mkOfCovers_I₀, Scheme.Hom.comp_app, IsIntegralHom.instIsStableUnderCompositionScheme, IsSeparated.isSeparated_eq_diagonal_isClosedImmersion, Scheme.Cover.isSheafFor_sigma_iff, Scheme.IdealSheafData.range_glueDataObjι_ι_eq_support_inter, Scheme.Pullback.gluing_t, Scheme.forgetToTop_comp_forget, Scheme.Pullback.left_affine_comp_pullback_hasPullback, Scheme.hom_base_inv_base, HasAffineProperty.affineAnd_isStableUnderComposition, IsLocalAtSource.iff_of_openCover, Scheme.GlueData.instHasMulticoequalizerDiagram, continuousMapPresheaf_map, Scheme.forgetToLocallyRingedSpace_map, IsOpenImmersion.range_pullback_fst_of_right, Proj.ι_comp_map, Scheme.toSpecΓ_naturality, WeaklyEtale.instIsStableUnderBaseChangeScheme, IsSeparated.diagonal_isClosedImmersion, pointEquivClosedPoint_apply_coe, instIsStableUnderBaseChangeSchemeLocallyQuasiFinite, Scheme.Hom.ι_toNormalization, geometrically_eq_universally, Scheme.Cover.ColimitGluingData.pullbackGluedIso_inv_fst_assoc, instGeometricallyIntegralSndScheme, Scheme.Cover.pullbackCoverOverProp_f, Spec.faithful, Spec.preimage_id, Proj.pullbackAwayιIso_hom_SpecMap_awayMap_right_assoc, isSheaf_propQCTopology_iff, Scheme.homeoOfIso_symm, Scheme.exists_isAffine_of_isLimit, Scheme.Cover.ColimitGluingData.transitionCocone_pt, Scheme.comp_coeBase_assoc, instCompactSpaceCarrierCarrierCommRingCatPullbackSchemeOfQuasiCompact_1, Scheme.comp_appLE, Surjective.instIsStableUnderBaseChangeScheme, IsLocalIso.le_of_isZariskiLocalAtSource, Scheme.Pullback.cocycle, IsFinite.instIsStableUnderBaseChangeScheme, Scheme.Pullback.pullbackP1Iso_hom_snd, IsProper.isStableUnderBaseChange, quasiSeparated_eq_diagonal_is_quasiCompact, IsOpenImmersion.hasLimit_cospan_forget_of_left, Scheme.IdealSheafData.inclusion_subschemeι, coprodMk_inl, IsClosedImmersion.respectsIso, Scheme.Opens.toSpecΓ_naturality_assoc, Scheme.GlueData.openCover_f, instSmoothOfRelativeDimensionOfNatNatCompScheme, Scheme.instHasIsosPrecoverageOfContainsIdentitiesOfRespectsIso, IsAffineOpen.isoSpec_inv, instIsMultiplicativeSchemeSurjective, Scheme.Hom.id_app, Scheme.Γ_obj, Scheme.Opens.fromSpecStalkOfMem_ι_assoc, instIsAffineXSchemeAffineCover, isLocallyNoetherian_iff_of_affine_openCover, Scheme.kerAdjunction_unit_app, Scheme.Hom.singleton_mem_qcPrecoverage, Scheme.Hom.liftQuotient_comp, Scheme.Pullback.carrierEquiv_symm_fst, Scheme.Hom.isPullback_resLE, Scheme.ProEt.forget_obj, Scheme.etalePrecoverage_le_proetalePrecoverage, instIsClosedImmersionLeftSchemeDiscretePUnitOneOverSpecOf, Scheme.Cover.ColimitGluingData.transitionMap_id, Proj.pullbackAwayιIso_inv_snd_assoc, Scheme.Hom.asFiberHom_fiberι_assoc, instIsClosedUnderIsomorphismsSchemeIrreducibleSpaceCarrierCarrierCommRingCat, instIsAffineHomDescScheme, Scheme.IdealSheafData.comapIso_hom_snd_assoc, Scheme.Hom.cover_X, Scheme.directedAffineCover_f, IsImmersion.isPullback_toImage_liftCoborder, Scheme.pullback_map_isOpenImmersion, Flat.instFstScheme, instIsOverTerminalScheme, Scheme.Opens.toSpecΓ_SpecMap_presheaf_map_assoc, Scheme.IdealSheafData.comap_id, Scheme.Pullback.range_fst_comp, instDescendsAlongSchemeMinMorphismPropertySurjectiveFlatLocallyOfFinitePresentationOfQuasiCompactOfIsZariskiLocalAtTarget, ΓSpec.adjunction_counit_app', quasiCompact_isStableUnderBaseChange, Proj.SpecMap_awayMap_awayι_assoc, Scheme.isoSpec_hom_naturality_assoc, pullbackSpecIso_inv_snd, IsImmersion.instMapDescScheme, Scheme.isoSpec_inv_image_zeroLocus, Scheme.IdealSheafData.range_glueDataObjι_ι, instPreservesColimitsOfShapeSchemeTopCatDiscreteForgetToTopOfSmall, HasAffineProperty.affineAnd_eq_of_propertyIsLocal, quasiCompact_comp, SurjectiveOnStalks.mono_of_injective, Scheme.Pullback.t'_fst_fst_snd_assoc, Etale.instSndScheme, Etale.instHasOfPostcompPropertySchemeMinMorphismPropertyLocallyOfFiniteTypeFormallyUnramified, IsOpenImmersion.isIso, Scheme.Cover.gluedCover_V, Proj.pullbackAwayιIso_inv_fst, isIso_morphismRestrict_iff_isIso_app, Scheme.hom_base_inv_base_assoc, Proj.awayι_comp_map_assoc, isPullback_inl_inl_coprodMap, IsFinite.eq_proper_inf_locallyQuasiFinite, SpecMap_ΓSpecIso_inv_toSpecΓ, AffineSpace.map_comp, Scheme.over_def, Proj.map_comp, HasAffineProperty.iff_of_openCover, Scheme.Hom.quasiFiniteAt_comp_iff_of_isOpenImmersion, IsOpenImmersion.instπWalkingCospanSchemeCospanOne, Proj.basicOpenToSpec_SpecMap_awayMap, isBasis_preimage_isAffineOpen, Scheme.Cover.coconeOfLocallyDirected_ι, Flat.instIsMultiplicativeScheme, Scheme.Cover.ι_fromGlued_assoc, universallyInjective_iff, instHasFiniteLimitsScheme, Scheme.GlueData.glue_condition, Scheme.GlueData.f_open, Scheme.OpenCover.isOpenCover_opensRange, pullbackRestrictIsoRestrict_inv_fst, Scheme.Cover.ulift_X, QuasiCompactCover.instCoverOfIsAffineOfFiniteI₀, Scheme.Cover.RelativeGluingData.ι_toBase, Scheme.openCoverOfIsOpenCover_f, instIsAffineCoprodScheme, Scheme.Pullback.Triplet.Spec_map_tensor_isPullback, IsOpenImmersion.isoOfRangeEq_inv_fac_assoc, Scheme.id_appTop, Scheme.Cover.gluedCoverT'_fst_fst, pullbackSpecIso_hom_base, Scheme.Cover.covers, IsOpenImmersion.comp, Scheme.instIsContinuousTopCatForgetToTopZariskiTopologyGrothendieckTopology, Scheme.Pullback.pullbackFstιToV_fst_assoc, smooth_comp, Scheme.Pullback.gluedLift_p2, ValuativeCriterion.Existence.eq, Scheme.IsLocallyDirected.openCover_f, Scheme.isCommMonObj_asOver_pullback, QuasiCompactCover.exists_hom, Scheme.GlueData.isOpen_iff, Surjective.sigmaDesc_of_union_range_eq_univ, QuasiCompactCover.isCompactOpenCovered_of_isCompact, Scheme.fromSpecStalk_toSpecΓ, IsOpenImmersion.instPreservesLimitSchemeLocallyRingedSpaceWalkingCospanCospanForgetToLocallyRingedSpace, WeaklyEtale.flat_diagonal, instRespectsIsoSchemeTopologicallyIsOpenEmbedding, Scheme.isMonHom_fst_id_right, IsLocalIso.eq_iInf, ValuativeCriterion.Existence.stableUnderBaseChange, Scheme.Spec_map_stalkMap_fromSpecStalk_assoc, Scheme.PartialMap.equiv_toPartialMap_iff_of_isSeparated, Etale.instIsMultiplicativeScheme, isInitial_iff_isEmpty, AffineSpace.reindex_over, Scheme.IdealSheafData.subschemeCover_map_subschemeι, Scheme.Cover.gluedCoverT'_snd_fst, AffineSpace.map_over, Scheme.isAffine_affineBasisCover, Scheme.Opens.toSpecΓ_SpecMap_presheaf_map_top_assoc, ExistsHomHomCompEqCompAux.exists_index, topologically_iso_le, Scheme.Cover.trans_id, sourceLocalClosure.property_coverMap_comp, Scheme.Cover.exists_lift_trans_eq, Scheme.Cover.exists_of_f_eq_f, Scheme.Hom.toImage_imageι_assoc, instIsStableUnderCompositionSchemeLocallyQuasiFinite, QuasiSeparated.diagonalQuasiCompact, Scheme.GlueData.instPreservesColimitTopCatWalkingMultispanProdJMultispanDiagramForgetToTop, ΓSpec.adjunction_counit_app, instRespectsIsoSchemeTopologicallyIsClosedEmbedding, instGeometricallyReducedFstScheme, Scheme.Hom.preimage_smoothLocus_eq, Scheme.coe_homeoOfIso_symm, Spec.map_id, instIrreducibleSpaceCarrierCarrierCommRingCatPullbackSchemeOfGeometricallyIrreducibleOfUniversallyOpen, Scheme.stalkMap_hom_inv_assoc, Scheme.Pullback.t_snd, universallyInjective_eq, instHasOfPostcompPropertySchemeQuasiSeparatedTopMorphismProperty, Proj.basicOpenIsoSpec_inv_ι, Scheme.id.base, Scheme.Γ_def, Scheme.Hom.isoOpensRange_hom_ι_assoc, instFaithfulOverSchemePullbackOfSurjectiveOfFlatOfQuasiCompact, AffineTargetMorphismProperty.respectsIso_mk, Spec.full, etale_le_weaklyEtale, Scheme.Pullback.Triplet.specTensorTo_snd, Scheme.directedAffineCover_I₀, opensDiagram_map, Scheme.Hom.isoOpensRange_hom_ι, Scheme.Cover.RelativeGluingData.ι_toBase_assoc, instHasOfPrecompPropertySchemeSurjective, morphismRestrict_ι, Scheme.Hom.comp_appLE, Scheme.Modules.restrictFunctorComp_hom_app_app, Scheme.Hom.preimageIso_inv_ι, Scheme.Pullback.gluing_J, Scheme.SpecMap_stalkSpecializes_fromSpecStalk, Scheme.descResidueField_fromSpecResidueField, Scheme.Cover.RelativeGluingData.equifibered, Scheme.Pullback.gluing_f, Scheme.Pullback.pullbackP1Iso_hom_fst_assoc, Spec.map_preimage_unop, Scheme.IdealSheafData.map_comp, instIsClosedImmersionSndScheme, WeaklyEtale.instCompScheme, Scheme.instEpiTopCatPullbackComparisonForgetToTop, Scheme.Modules.restrictFunctorComp_inv_app_app, Scheme.Cover.instIsLocallyDirectedI₀CompFunctorOfLocallyDirectedForget, IsOpenImmersion.isPullback_lift_id, IsLocalIso.le_of_isLocalAtSource, Proj.pullbackAwayιIso_inv_snd, Scheme.SpecΓIdentity_app, Proj.pullbackAwayιIso_inv_fst_assoc, Scheme.IdealSheafData.inclusion_comp_assoc, Scheme.homeoOfIso_apply, pullbackSpecIso_inv_fst'_assoc, affineLocally_respectsIso, Scheme.LocalRepresentability.comp_toGlued_eq, IsZariskiLocalAtSource.sigmaDesc, ExistsHomHomCompEqCompAux.h𝒰X, isSheaf_type_propQCTopology_iff, isIso_stalkMap_coprodSpec, Proj.pullbackAwayιIso_hom_awayι, instPreservesColimitsOfShapeOppositeCommRingCatSchemeDiscreteSpecOfFinite, Scheme.IsLocallyDirected.homOfLE_tAux, isOpenImmersion_stableUnderBaseChange, Scheme.Opens.toSpecΓ_SpecMap_map_assoc, hasOfPostcompProperty_isOpenImmersion_of_morphismRestrict, IsImmersion.instDiagonalScheme, exists_appTop_π_eq_of_isAffine_of_isLimit, Scheme.IdealSheafData.ker_fst_of_isClosedImmersion, Scheme.GlueData.instPreservesColimitWalkingMultispanProdJMultispanDiagramForget, Scheme.Pullback.pullbackP1Iso_hom_ι_assoc, specializingMap_respectsIso, Scheme.OpenCover.restrict_f, Scheme.Pullback.pullbackFstιToV_fst, Scheme.instIsOverMapResidueFieldMapOverInferInstanceOverClass, sourceLocalClosure.instRespectsIsoScheme, WeaklyEtale.instRespectsIsoScheme, Scheme.OpenCover.iSup_opensRange, Scheme.GlueData.ι_isOpenImmersion, Scheme.Pullback.Triplet.specTensorTo_snd_assoc, Scheme.Cover.mkOfCovers_X, IsOpenImmersion.instPreservesLimitSchemeTopCatWalkingCospanCospanForgetToTop_1, IsOpenImmersion.hasLimit_cospan_forget_of_right, universallyClosed_isStableUnderBaseChange, Scheme.fppfPrecoverage_le_fpqcPrecoverage, Scheme.Pullback.ofPointTensor_SpecTensorTo_assoc, FormallyUnramified.isOpenImmersion_diagonal, AffineTargetMorphismProperty.arrow_mk_iso_iff, smooth_isStableUnderBaseChange, IsFinite.instCompScheme, Scheme.Pullback.pullbackP1Iso_inv_snd_assoc, Scheme.kerAdjunction_counit_app, Scheme.Pullback.t_fst_fst, opensDiagramι_app, Scheme.Pullback.carrierEquiv_symm_snd, IsClosedImmersion.comp_iff, Proj.pullbackAwayιIso_hom_SpecMap_awayMap_right, instIsOpenImmersionCompSchemeιQuasiFiniteLocusToNormalization, Scheme.zariskiPrecoverage_le_fppfPrecoverage, isCompl_range_inl_inr, UniversallyClosed.out, Scheme.exists_cover_of_mem_pretopology, universallyClosed_isStableUnderComposition, Scheme.Cover.ColimitGluingData.fst_gluedCocone_ι, ΓSpec.adjunction_unit_app, Scheme.Opens.toSpecΓ_top, universally_isLocalAtTarget, Scheme.LocalRepresentability.glueData_openCover_map, HasRingHomProperty.stableUnderComposition, HasRingHomProperty.isMultiplicative, Scheme.Pullback.range_map, Scheme.ProEt.instLocallyCoverDenseOverForgetOverProetaleTopology, Scheme.Pullback.openCoverOfRight_f, Scheme.Cover.forgetQc_toPreZeroHypercover, Scheme.Pullback.t'_snd_snd, SurjectiveOnStalks.isEmbedding_pullback, AffineSpace.SpecIso_inv_appTop_coord, IsOpenImmersion.to_iso, AffineSpace.functor_obj_map, AffineSpace.instLocallyOfFinitePresentationOverSchemeInferInstanceOverClassOfFinite, instSurjectiveCompScheme, Scheme.Hom.iInf_ker_openCover_map_comp, Scheme.AffineZariskiSite.restrictIsoSpec_hom_app, specTargetImageFactorization_comp_assoc, instIsIsoSchemeMorphismRestrict, Scheme.Cover.LocallyDirected.ofIsBasisOpensRange_trans, QuasiCompactCover.of_isOpenMap, AffineSpace.map_reindex_assoc, instGeometricallyIrreducibleSndScheme, instIsMultiplicativeSchemeQuasiSeparated, IsLocalAtSource.comp, instRespectsIsoSchemeSurjective, instSubsingletonOverTerminalScheme, Scheme.AffineCover.cover_f, instIsOpenImmersionInrScheme, instHasInitialScheme, IsLocalIso.instIsStableUnderBaseChangeScheme, SurjectiveOnStalks.comp, Scheme.Cover.gluedCover_J, Scheme.stalkMap_congr_hom_assoc, Scheme.Hom.opensRange_pullbackFst, Scheme.Cover.ColimitGluingData.transitionCocone_ι_app, pullbackRestrictIsoRestrict_hom_ι_assoc, exists_map_preimage_eq_map_preimage, Scheme.Cover.sigmaFunctor_obj, compactSpace_iff_quasiCompact, Scheme.mem_overGrothendieckTopology, UniversallyOpen.snd, Scheme.Modules.pseudofunctor_right_unitality, Scheme.Modules.pseudofunctor_mapComp_hom_τl, sourceLocalClosure.instIsStableUnderCompositionSchemeOfIsStableUnderBaseChange, IsAffineOpen.isoSpec_inv_toSpecΓ, disjoint_opensRange_sigmaι, Scheme.comp_base_assoc, Scheme.stalkMap_id, isIso_iff_isOpenImmersion_and_surjective, instIsAffineHomMapOverSchemeOpensDiagram, IsAffineOpen.toSpecΓ_isoSpec_inv, Scheme.IsLocallyDirected.fst_inv_eq_snd_inv, AffineSpace.instIsIsoSchemeOverInferInstanceOverClassOfIsEmpty, Scheme.Hom.instIsIsoNormalizationPullbackOfSmooth, Scheme.Opens.toSpecΓ_SpecMap_presheaf_map, quasiCompact_isStableUnderComposition, Scheme.Hom.comp_app_assoc, Scheme.comp_app, instIsStableUnderCompositionSchemeLocallyOfFinitePresentation, isIso_iff_isOpenImmersion, isOpenImmersion_isMultiplicative, AffineSpace.homOfVector_toSpecMvPoly_assoc, exists_preimage_eq, Scheme.Cover.glued_cover_cocycle_snd, Scheme.Hom.normalizationPullback_snd_assoc, Scheme.GlueData.ι_eq_iff, Scheme.Cover.trans_map, AffineSpace.comp_homOfVector, Scheme.openCoverOfIsOpenCover_X, ι_left_coprodIsoSigma_inv, instHasStrictInitialObjectsScheme, AffineSpace.isOpenMap_over, instIsLocallyNoetherianPullbackSchemeOfLocallyOfFiniteType_1, Scheme.isoSpec_inv_naturality_assoc, Spec.map_comp_assoc, IsImmersion.isImmersion_iff_exists, Scheme.homOfLE_homOfLE_assoc, IsLocalAtSource.respectsLeft_isOpenImmersion, instPreservesColimitsOfShapeOppositeCommRingCatSchemeDiscreteWalkingPairSpec, Scheme.Pullback.pullbackP1Iso_inv_fst, Proj.map_id, isLocalIso_iff, Spec.preimage_comp_assoc, Scheme.Cover.toSigma_s₀, IsClosedImmersion.lift_fac, AffineSpace.hom_ext_iff, Scheme.SpecMap_stalkMap_fromSpecStalk_assoc, Scheme.OpenCover.pullbackCoverAffineRefinementObjIso_inv_pullbackHom_assoc, instHasOfPostcompPropertySchemeQuasiCompactQuasiSeparated, Scheme.Pullback.pullbackP1Iso_inv_fst_assoc, Scheme.singleton_mem_precoverage_iff, Scheme.IdealSheafData.glueDataObjMap_glueDataObjι_assoc, QuasiCompactCover.inst, Scheme.ProEt.topology_eq_inducedTopology, Scheme.Pullback.tensorCongr_SpecTensorTo_assoc, QuasiCompactCover.instPullback₁Scheme, Scheme.isoSpec_Spec_hom, AffineScheme.forgetToScheme_full, Scheme.IdealSheafData.comapIso_inv_subschemeι_assoc, IsZariskiLocalAtTarget.coprodMap, IsFinite.instDescScheme, AffineSpace.map_toSpecMvPoly, Proj.pullbackAwayιIso_hom_SpecMap_awayMap_left, instHasFiniteCoproductsScheme, AffineSpace.isPullback_map, Scheme.Hom.inl_normalizationCoprodIso_hom_fromNormalization_assoc, exists_appTop_map_eq_zero_of_isAffine_of_isLimit, instIsMultiplicativeSchemeUniversallyInjective, nonempty_isColimit_cofanMk_of, Scheme.Cover.pullbackCoverOverProp'_X, AffineScheme.forgetToScheme_faithful, instQuasiCompactSndScheme, pullbackSpecIso_inv_fst, Scheme.Pullback.t'_snd_fst_fst, isIso_of_comp_eq_sigmaSpec, IsLocalAtTarget.descendsAlong, UniversallyInjective.respectsIso, HasAffineProperty.coprodDesc_affineAnd, IsFinite.instFstScheme, Scheme.zariskiPrecoverage_le_propQCPrecoverage, Scheme.Pullback.affine_affine_hasPullback, Scheme.grothendieckTopology_eq_inf, AffineSpace.homOfVector_over_assoc, stalkwise_respectsIso, Scheme.GlueData.openCover_X, morphismRestrict_comp, Scheme.SpecMap_stalkSpecializes_fromSpecStalk_assoc, Scheme.Hom.comp_appLE_assoc, Scheme.isPullback_toSpecΓ_toSpecΓ, Scheme.Cover.LocallyDirected.trans_comp, Scheme.Hom.stalkMap_hom_inv, instIsMultiplicativeSchemeLocallyOfFiniteType, Scheme.Cover.pullbackCoverOverProp'_f, Scheme.restrictFunctorΓ_hom_app, Scheme.Cover.functorOfLocallyDirected_map, Scheme.stalkMap_congr_hom, Scheme.instEffectiveEpiOfQuasiCompactOfSurjectiveOfFlat, Scheme.Modules.pseudofunctor_mapComp_hom_τr, Scheme.Γ_map_op, Scheme.Pullback.range_snd_comp, Scheme.directedAffineCover_X, Scheme.Cover.property_trans, instIsIsoSchemeAppUnitOppositeCommRingCatAdjunctionOfIsAffine, Scheme.Modules.restrictFunctorId_inv_app_app, Scheme.Opens.isoOfLE_hom_ι, IsClosedImmersion.isStableUnderBaseChange, Scheme.Opens.over_def, Scheme.Opens.fromSpecStalkOfMem_ι, Scheme.Modules.pushforwardComp_hom_app_app, Scheme.Cover.ColimitGluingData.fst_gluedCocone_ι_assoc, Scheme.iso_inv_base_hom_base, Scheme.Hom.inl_normalizationCoprodIso_hom_fromNormalization, Scheme.toSpecΓ_isoSpec_inv, Scheme.Hom.singleton_mem_fppfPrecoverage, SpecToEquivOfLocalRing_apply_fst, Scheme.Hom.comp_apply, Scheme.Cover.sigma_X, Scheme.homOfLE_homOfLE, Scheme.smallGrothendieckTopologyOfLE_eq_toGrothendieck_smallPretopology, Scheme.ProEt.forget_map, Proj.awayι_toSpecZero, opensDiagram_obj, Scheme.Pullback.gluing_U, WeaklyEtale.instIsMultiplicativeScheme, Scheme.ProEt.mk_hom, Scheme.Pullback.t'_fst_fst_fst, Scheme.Cover.nonempty_of_nonempty, Scheme.instSmallPrecoverage, Scheme.ProEt.instFaithfulOverForget, IsProper.stableUnderComposition, Scheme.Pullback.t_id, Scheme.toIso_inv_ι_assoc, Scheme.instFullOppositeIdealSheafDataOverSubschemeFunctor, isPullback_SpecMap_of_isPushout, instIsOpenImmersionMapScheme, Scheme.Hom.cover_f, Scheme.presieve₀_mem_qcPrecoverage_iff, sigmaι_eq_iff, opensCone_π_app, instHasColimitOverScheme, Scheme.Hom.resLE_id, isOpenImmersion_respectsIso, Scheme.isClosedUnderIsomorphisms_quasiCompactCover, Scheme.stalkMap_comp, Scheme.AffineZariskiSite.directedCover_X, IsPreimmersion.instMonoScheme, Scheme.Cover.ulift_f, Scheme.IdealSheafData.comapIso_inv_subschemeι, UniversallyOpen.fst, instIsRightAdjointCommRingCatOppositeSchemeΓ, Scheme.OpenCover.instIsOpenImmersionTrans, Scheme.IdealSheafData.glueDataObjMap_glueDataObjι, Scheme.Hom.comp_appIso, isIso_of_isEmpty, instIsStableUnderBaseChangeSchemeGeometricallyIntegral, IsOpenImmersion.range_pullback_to_base_of_left, instSurjectiveDescI₀SchemeF, over_def, IsIntegralHom.instFstScheme, Proj.homOfLE_toBasicOpenOfGlobalSections_ι_assoc, Scheme.Cover.ColimitGluingData.cocone_ι_transitionMap_assoc, instLocallyOfFinitePresentationFstScheme, Scheme.Hom.stalkMap_inv_hom_assoc, WeaklyEtale.instHasOfPostcompPropertyScheme, isAffineHom_isStableUnderBaseChange, Scheme.Opens.toSpecΓ_SpecMap_appLE_assoc, Scheme.IdealSheafData.inclusion_comp, Scheme.Γ_map, Scheme.Cover.pullbackCoverOver'_I₀, AffineScheme.forgetToScheme_map, Scheme.Cover.ulift_I₀, IsIntegralHom.instIsMultiplicativeScheme_1, Scheme.LocalRepresentability.yoneda_toGlued_yonedaGluedToSheaf_assoc, isSeparated_iff, Proj.awayι_toSpecZero_assoc, Scheme.Hom.inr_normalizationCoprodIso_hom_fromNormalization_assoc, HasRingHomProperty.inf, Scheme.instSmallPropQCPrecoverage, IsClosedImmersion.instIsMultiplicativeScheme, IsOpenImmersion.lift_fac_assoc, Scheme.Cover.mem_propQCTopology, instIsIntegralPullbackSchemeOfGeometricallyIntegralOfFlatOfUniversallyOpenOfIsLocallyNoetherian_1, isSheaf_fpqcTopology_continuousMapPresheafAb, ValuativeCommSq.commSq, Scheme.residueFieldCongr_fromSpecResidueField, universallyInjective_eq_diagonal, Scheme.coprodPresheafObjIso_hom_snd, IsOpenImmersion.instHasOfPostcompPropertyScheme, coprodSpec_coprodMk, Scheme.Cover.exists_of_trans_eq_trans, Scheme.instIsStableUnderBaseChangeJointlySurjectivePrecoverage, isAffine_affineScheme, Scheme.Hom.preimageIso_hom_ι, IsClosedImmersion.iff_isFinite_and_mono, Scheme.Cover.sigmaFunctor_map, Scheme.Pullback.cocycle_fst_fst_snd, UniversallyOpen.instIsStableUnderCompositionSchemeTopologicallyIsOpenMap, Scheme.isoOfEq_hom_ι_assoc, isIso_iff_isOpenImmersion_and_epi_base, instRespectsSchemeLocallyQuasiFiniteIsOpenImmersion, Scheme.isoSpec_inv_naturality, IsAffineOpen.toSpecΓ_fromSpec_assoc, Scheme.Hom.ker_comp_of_isIso, Scheme.Pullback.openCoverOfRight_X, Scheme.instSubcanonicalFpqcTopology, instMonoObjWalkingSpanCompSchemeSpanForgetNoneWalkingPairSomeMapInitOfIsOpenImmersion, Scheme.Hom.id_appIso, Scheme.isoOfEq_inv_ι_assoc, Scheme.Hom.preimageIso_inv_ι_assoc, instQuasiSeparatedSndScheme, Scheme.Modules.pseudofunctor_mapId_hom_τr, Scheme.Cover.gluedCoverT'_snd_fst_assoc, Scheme.Hom.asFiberHom_fiberι, Scheme.IdealSheafData.subschemeFunctor_obj, specTargetImageFactorization_comp, Scheme.comp_appTop_assoc, universallyInjective_isStableUnderComposition, instHasColimitsOfShapeDiscreteSchemeOfSmall, IsFinite.instSndScheme, Scheme.Cover.ι_glueMorphisms, Proj.pullbackAwayιIso_hom_SpecMap_awayMap_left_assoc, Scheme.Hom.resLE_comp_resLE_assoc, IsZariskiLocalAtSource.respectsLeft_isOpenImmersion, Scheme.AffineZariskiSite.directedCover_f, IsImmersion.instIsMultiplicativeScheme, instIsOpenImmersionMapSchemeCompOverOverTopMorphismPropertyForgetForget, instIsAffineObjOppositeCommRingCatSchemeSpec, AffineSpace.reindex_comp, Scheme.IsLocallyDirected.ι_eq_ι_iff, AffineSpace.map_SpecMap, emptyIsInitial_to, instLocallyOfFiniteTypeSndScheme, Scheme.isoSpec_Spec, Scheme.IsLocallyDirected.openCover_I₀, Scheme.Hom.normalizationDesc_comp, Scheme.GlueData.oneHypercover_p₂, instGeometricallyIntegralFstScheme, isOpenImmersion_eq_inf, Scheme.Pullback.Triplet.specTensorTo_base_fst, compactSpace_iff_exists, Scheme.LocalRepresentability.glueData_V, Surjective.comp_iff, instHasCoproductsOfShapeOverSchemeTopMorphismPropertyOfSmall, Scheme.IsLocallyDirected.instPreservesColimitLocallyRingedSpaceForgetToLocallyRingedSpace, Proj.fromOfGlobalSections_toSpecZero, Scheme.Cover.ι_glueMorphisms_assoc, Scheme.iso_hom_base_inv_base, instIsOpenImmersionLeftSchemeDiscretePUnitιOverTopMorphismProperty, AffineSpace.SpecIso_inv_over_assoc, sourceLocalClosure.instContainsIdentitiesScheme, Scheme.Hom.singleton_mem_propQCPrecoverage, AffineSpace.instIsAffineHomOverSchemeInferInstanceOverClass, sourceLocalClosure.instRespectsLeftSchemeOfIsStableUnderBaseChange, Scheme.stalkMap_hom_inv, IsClosedImmersion.eq_inf, sourceAffineLocally_respectsIso, Scheme.instIsIsoIrreducibleComponentιOfIrreducibleSpaceCarrierCarrierCommRingCat, Scheme.monObjAsOverPullback_mul, isClosedMap_isStableUnderComposition, Scheme.ProEt.instPreservesFiniteLimitsOverForget, HasAffineProperty.instRespectsIsoScheme, toSpecΓ_SpecMap_ΓSpecIso_inv_assoc, instHasTerminalScheme, Scheme.Modules.pseudofunctor_obj_obj, IsClosedImmersion.isIso_lift, targetAffineLocally_affineAnd_eq_affineLocally, Scheme.Hom.comp_preimage, Scheme.kerFunctor_map, IsZariskiLocalAtSource.iff_of_openCover, Scheme.stalkMap_hom_inv_apply, AffineSpace.reindex_comp_assoc, Scheme.Hom.iInf_ker_openCover_map_comp_apply, Surjective.instSndScheme, QuasiCompactCover.instBindScheme, Scheme.Hom.normalizationCoprodIso_inv_coprodDesc_fromNormalization, pullbackSpecIso_inv_snd_assoc, AffineSpace.instIsOverHomOfVectorOverSchemeInferInstanceOverClass, Scheme.Cover.pushforwardIso_f, Scheme.isoOfEq_rfl, Scheme.inv_appTop, isPullback_Spec_map_pushout, Scheme.GlueData.sheafValGluedMk_val, IsIntegralHom.instHasOfPostcompPropertySchemeIsSeparated, Scheme.Hom.quasiFiniteAt_iff, IsProper.comp_iff, instIsOpenImmersionInlScheme, Scheme.LocalRepresentability.instIsIsoSheafZariskiTopologyTypeYonedaGluedToSheaf, Scheme.LocalRepresentability.glueData_f, universallyClosed_fst, Scheme.IsLocallyDirected.openCover_X, Scheme.Spec_map_stalkSpecializes_fromSpecStalk_assoc, instIsLocallyNoetherianPullbackSchemeOfLocallyOfFiniteType, Scheme.Hom.exists_mem_and_isIso_morphismRestrict_toNormalization, AffineTargetMorphismProperty.cancel_right_of_respectsIso, Scheme.Cover.copy_I₀, HasAffineProperty.descendsAlong_of_affineAnd, Scheme.fullyFaithfulForgetToLocallyRingedSpace_preimage_toLRSHom, Scheme.LocalRepresentability.glueData_t', IsOpenImmersion.hasPullback_of_right, Scheme.Cover.RelativeGluingData.cover_f, Scheme.instIsOverFstOverInferInstanceOverClassId, IsImmersion.instFstScheme, AffineSpace.toSpecMvPolyIntEquiv_apply, Scheme.Cover.pullbackCoverOver_f, Scheme.descResidueField_stalkClosedPointTo_fromSpecResidueField, Scheme.Pullback.openCoverOfBase_X, isOpenImmersion_sigmaDesc, IsOpenImmersion.isoOfRangeEq_hom_fac, coprodSpec_inr_assoc, Scheme.instIsStableUnderSupQcPrecoverage, AffineSpace.functor_obj_obj, Scheme.Spec.map_residueFieldIso_inv_eq_fromSpecResidueField, IsClosedImmersion.Spec_iff, Scheme.Pullback.pullbackP1Iso_hom_snd_assoc, Scheme.IsLocallyDirected.instIsLocallyDirectedWidePushoutShapeCompForgetOfIsOpenImmersionMap, Scheme.Cover.toPresieveOver_le_arrows_iff, isPreimmersion_eq_inf, Scheme.Hom.comp_toLRSHom, Scheme.Pullback.cocycle_fst_fst_fst, AffineSpace.map_id, Scheme.Pullback.openCoverOfBase'_f, instIsOpenImmersionMapWalkingSpanSchemeSpan, AffineSpace.isIntegralHom_over_iff_isEmpty, pullbackSpecIso_inv_fst_assoc, Scheme.Pullback.ofPointTensor_SpecTensorTo, WeaklyEtale.instSndScheme, Scheme.Hom.id_base, AffineSpace.functor_map_app, UniversallyOpen.eq, Scheme.LocalRepresentability.yoneda_toGlued_yonedaGluedToSheaf, IsClosedImmersion.isIso_of_ker_eq, IsOpenImmersion.range_pullback_snd_of_left, instIsMultiplicativeSchemeLocallyQuasiFinite, Scheme.instIsIsoSubschemeιBotIdealSheafData, Scheme.Cover.gluedCover_f, isProper_eq, ΓSpec.adjunction_homEquiv_apply, UniversallyOpen.universally_isOpenMap, Scheme.Pullback.t_fst_snd, Scheme.PartialMap.ofFromSpecStalk_comp, Scheme.PartialMap.isOver_iff_eq_restrict, Scheme.Opens.isoOfLE_inv_ι, Scheme.Hom.ι_toNormalization_assoc, instQuasiCompactLiftSchemeIdOfQuasiSeparatedSpaceCarrierCarrierCommRingCat, Scheme.Hom.Spec_map_residueFieldMap_fromSpecResidueField_assoc, Scheme.ProEt.instWeaklyEtaleHomDiscretePUnit, Flat.comp, Scheme.Cover.Hom.sigma_h₀, Scheme.Cover.overEquiv_generate_toPresieveOver_eq_ofArrows, Scheme.isoSpec_inv_toSpecΓ_assoc, Proj.fromOfGlobalSections_morphismRestrict, Scheme.Cover.LocallyDirected.trans_id, Scheme.Hom.isoImage_inv_ι, AffineSpace.map_comp_assoc, Scheme.isSeparated_iff_isClosedImmersion_prod_lift, Scheme.GlueData.ι_isoCarrier_inv, IsFinite.instIsMultiplicativeScheme, Scheme.IsLocallyDirected.glueDataι_naturality, Scheme.Cover.toSigma_h₀, universallyOpen_iff, Scheme.IdealSheafData.range_glueDataObjι, Scheme.Hom.isoImage_inv_ι_assoc, IsZariskiLocalAtTarget.descendsAlong_inf_quasiCompact, locallyOfFinitePresentation_isStableUnderBaseChange, Scheme.Modules.pushforwardId_hom_app_app, pullbackRestrictIsoRestrict_hom_ι, Scheme.Pullback.gluing_ι, HasRingHomProperty.descendsAlong, Scheme.Pullback.gluedLiftPullbackMap_fst_assoc, Scheme.Hom.ker_comp, Scheme.coprodPresheafObjIso_hom_fst, ExistsHomHomCompEqCompAux.ha, IsOpenImmersion.range_pullback_to_base_of_right, Scheme.IdealSheafData.comapIso_hom_snd, UniversallyClosed.universally_isClosedMap, IsLocalAtTarget.iff_of_openCover, isRegularEpi_of_flat_of_surjective_of_isAffine, Scheme.instHasIsosQcPrecoverage, smoothOfRelativeDimension_comp, Scheme.directedAffineCover_trans, Scheme.Pullback.t'_snd_fst_snd, instIsStableUnderCompositionSchemeLocallyOfFiniteType, Scheme.IsLocallyDirected.instIsOpenImmersionι, instQuasiCompactFstScheme, Scheme.coverOfIsIso_X, isCompl_opensRange_inl_inr, universallyClosed_eq_universallySpecializing, Scheme.Pullback.Triplet.fst_SpecTensorTo_apply, ΓSpecIso_inv_ΓSpec_adjunction_homEquiv, IsLocalIso.instIsMultiplicativeScheme, IsOpenImmersion.instPreservesLimitSchemeWalkingCospanCospanForget_1, Scheme.LocalRepresentability.instIsLocallyInjectiveHomYonedaGluedToSheaf, Scheme.Cover.LocallyDirected.property_trans, instHasOfPostcompPropertySchemeIsProperIsSeparated, Scheme.Cover.hasColimit_of_locallyDirected, instIsOpenImmersionSigmaSpec, topologically_respectsIso, ExistsHomHomCompEqCompAux.hab, Proj.basicOpenToSpec_SpecMap_awayMap_assoc, toSpecΓ_SpecMap_ΓSpecIso_inv, HasRingHomProperty.isStableUnderBaseChange, exists_map_preimage_le_map_preimage, AffineSpace.homOfVector_over, IsOpenImmersion.isPullback, Scheme.Hom.inv_appTop, Flat.surjective_descendsAlong_surjective_inf_flat_inf_quasicompact, Scheme.PartialMap.restrict_hom, FormallyUnramified.instIsMultiplicativeScheme, Scheme.AffineZariskiSite.opensRange_relativeGluingData_map, Scheme.Cover.ColimitGluingData.prop_trans, Scheme.Hom.toNormalization_inr_normalizationCoprodIso_hom_assoc, Scheme.qcCoverFamily_property, IsProper.instIsMultiplicativeScheme, pullbackSpecIso_hom_fst, IsProper.eq_valuativeCriterion, Scheme.Pullback.base_affine_hasPullback, locallyOfFinitePresentation_comp, Scheme.instIsStableUnderCompositionPrecoverageOfIsStableUnderComposition, IsClosedImmersion.eq_isFinite_inf_mono, quasiSeparated_iff, AffineSpace.reindex_over_assoc, Scheme.IdealSheafData.comapIso_hom_fst_assoc, Scheme.IsJointlySurjectivePreserving.exists_preimage_snd_triplet_of_prop, Scheme.Cover.pullbackCoverOverProp_X, QuasiCompactCover.instSumScheme_1, IsSeparated.stableUnderComposition, instIsAffineTerminalScheme, Spec.map_comp, Scheme.AffineZariskiSite.PreservesLocalization.isLocallyDirected, exists_app_map_eq_map_of_isLimit, Scheme.proetalePrecoverage_le_fpqcPrecoverage, instIsStableUnderBaseChangeSchemeGeometrically, Scheme.Pullback.cocycle_snd_fst_fst, Scheme.Pullback.pullbackP1Iso_inv_snd, isIso_of_isClosedImmersion_of_surjective, Scheme.isPullback_of_openCover, instIsMultiplicativeSchemeUniversallyClosed, Scheme.Cover.gluedCoverT'_fst_snd_assoc, instIsOpenImmersionAppOverSchemeOpensDiagramι, Scheme.Pullback.range_snd, Scheme.Hom.stalkMap_comp, Scheme.SpecΓIdentity_inv_app, Scheme.Pullback.diagonalCoverDiagonalRange_eq_top_of_injective, Scheme.Cover.coconeOfLocallyDirected_pt, instLocallyOfFinitePresentationSndScheme, UniversallyClosed.eq_valuativeCriterion, Scheme.Cover.exists_eq, preservesLimit_rightOp_Γ, Scheme.ofArrows_mem_precoverage_iff, isClosedImmersion_equalizer_ι_left, Scheme.Cover.ColimitGluingData.trans_app_left, instRespectsIsoSchemeTopologicallyInjective, IsSeparated.instCompScheme, FormallyUnramified.instIsStableUnderBaseChangeScheme, Spec.homEquiv_apply, Scheme.IsLocallyDirected.ι_jointly_surjective, Scheme.AffineCover.cover_X, Scheme.Pullback.t'_fst_fst_snd, instLocallyOfFiniteTypeFstScheme, Scheme.jointlySurjectiveTopology_eq_toGrothendieck_jointlySurjectivePretopology, AffineSpace.isoOfIsAffine_hom_appTop, Etale.etale_isStableUnderBaseChange, coprodSpec_inl_assoc, Scheme.Opens.fromSpecStalkOfMem_toSpecΓ, Proj.pullbackAwayιIso_hom_awayι_assoc, Scheme.instLocallyCoverDenseOverTopMorphismPropertyOverForgetOverGrothendieckTopology, Scheme.isoSpec_Spec_inv, Spec.map_eqToHom, Scheme.affineBasisCover_obj, Scheme.Hom.comp_toLRSHom_assoc, Scheme.Cover.ColimitGluingData.transitionMap_comp, HasAffineProperty.diagonal_of_openCover_diagonal, sigmaOpenCover_I₀, Scheme.Cover.Hom.sigma_s₀, isomorphisms_eq_isOpenImmersion_inf_surjective, Scheme.grothendieckTopology_monotone, Scheme.topIso_hom, instPreservesColimitsOfShapeOppositeCommRingCatSchemeDiscretePEmptySpec, Scheme.Hom.map_resLE, Scheme.Cover.sigma_I₀, Scheme.instEtaleF, IsPreimmersion.instSndScheme, Scheme.isSeparated_iff, Scheme.bot_mem_propQCPrecoverage, Scheme.inv_hom_apply, Proj.SpecMap_awayMap_awayι, basicOpenIsoSpecAway_inv_homOfLE_assoc, exists_mem_of_isClosed_of_nonempty', SurjectiveOnStalks.instIsMultiplicativeScheme, IsClosedImmersion.iff_isProper_and_mono, Scheme.instIsStableUnderBaseChangePrecoverageOfIsJointlySurjectivePreservingOfIsStableUnderBaseChange, Scheme.Spec.map_residueFieldIso_inv_eq_fromSpecResidueField_assoc, Scheme.Cover.pushforwardIso_I₀, sourceLocalClosure.instIsStableUnderBaseChangeScheme, Scheme.comp_toLRSHom, ι_sigmaSpec_assoc, instIrreducibleSpaceCarrierCarrierCommRingCatPullbackSchemeOfGeometricallyIrreducibleOfUniversallyOpen_1, Scheme.Pullback.cocycle_fst_snd, Scheme.IdealSheafData.subSchemeCover_map_inclusion_assoc, Scheme.Cover.ColimitGluingData.gluedCocone_pt, isPullback_morphismRestrict, instIsAffineXSchemeCoverOfIsIsoIsOpenImmersionId, isPullback_inr_inr_coprodMap, Scheme.OpenCover.finiteSubcover_X, instIsAffineHomCompScheme, Scheme.Pullback.range_fst, IsOpenImmersion.instPreservesLimitSchemeTopCatWalkingCospanCospanForgetToTop, instIsMultiplicativeSchemeIsDominant, Scheme.fppfTopology_le_fpqcTopology, instHasOfPostcompPropertySchemeIsSeparatedTopMorphismProperty, Scheme.fppfPrecoverage_eq_inf, instRespectsIsoSchemeTargetAffineLocallyOfToProperty, AffineSpace.isoOfIsAffine_hom, instIsClosedImmersionFstScheme, AffineTargetMorphismProperty.isStableUnderBaseChange_of_isStableUnderBaseChangeOnAffine_of_isZariskiLocalAtTarget, Scheme.Hom.resLE_comp_ι_assoc, smoothOfRelativeDimension_isStableUnderBaseChange, geometrically_inf, Scheme.Hom.ι_fromNormalization_assoc, Scheme.Cover.trans_comp, instMonoCoprodScheme, Scheme.emptyTo_c_app, Scheme.stalkMap_inv_hom_apply, basicOpenIsoSpecAway_inv_homOfLE, Scheme.Hom.map_resLE_assoc, IsLocalIso.exists_isOpenImmersion, Scheme.Hom.comp_base_assoc, Scheme.propQCPrecoverage_le_precoverage, Scheme.OpenCover.pullbackCoverAffineRefinementObjIso_inv_map, coprodSpec_inl, Scheme.Pullback.pullbackP1Iso_hom_ι, Scheme.GlueData.oneHypercover_X, IsImmersion.instHasOfPostcompPropertySchemeTopMorphismProperty, Scheme.IdealSheafData.subSchemeCover_map_inclusion, Scheme.Modules.conjugateEquiv_pullbackId_hom, QuasiCompactCover.singleton, Scheme.Pullback.range_diagonal_subset_diagonalCoverDiagonalRange, ΓSpec.adjunction_homEquiv_symm_apply, Scheme.Pullback.openCoverOfLeftRight_I₀, Scheme.Hom.toNormalization_normalizationDesc_assoc, IsAffineOpen.isoSpec_hom_apply, Scheme.Modules.conjugateEquiv_pullbackComp_inv, Scheme.smallGrothendieckTopology_eq_toGrothendieck_smallPretopology, Scheme.zariskiPrecoverage_le_fpqcPrecoverage, IsFinite.eq_isProper_inf_isAffineHom, quasiSeparatedSpace_iff_quasiSeparated, isPullback_of_isClosedImmersion, Scheme.Hom.resLE_comp_ι, exists_appTop_π_eq_of_isLimit, Scheme.Hom.liftCoborder_ι_assoc, targetAffineLocally_affineAnd_le, IsAffine.affine, Scheme.Pullback.Triplet.Spec_ofPointTensor_SpecTensorTo, Scheme.Pullback.t'_snd_snd_assoc, Scheme.IdealSheafData.inclusion_id_assoc, Scheme.isIso_subschemeι_iff_eq_bot, instIsMultiplicativeSchemeSmoothOfRelativeDimensionOfNatNat, Etale.instHasOfPostcompPropertyScheme, Scheme.Pullback.Triplet.ofPoint_s, QuasiCompactCover.isCompactOpenCovered_of_isAffineOpen, Scheme.coverOfIsIso_f, Spec.map_inv, Scheme.Pullback.instHasPullback, Scheme.Modules.pseudofunctor_left_unitality, Scheme.Hom.isoOpensRange_inv_comp_assoc, Scheme.Modules.pseudofunctor_associativity_assoc, Scheme.instWeaklyEtaleF, Scheme.instHasPullbacksIsOpenImmersion, Scheme.Hom.cover_I₀, Scheme.exists_isQuasiAffine_of_isLimit, pullbackSpecIso_hom_base_assoc, IsAffineOpen.isoSpec_inv_toSpecΓ_assoc, Scheme.precoverage_le_qcPrecoverage_of_isOpenMap, Scheme.Cover.gluedCoverT'_snd_snd, universally_isZariskiLocalAtSource, instIsLocallyDirectedCompSchemeOverOverTopMorphismPropertyForgetForgetForget, isPullback_SpecMap_pushout, Scheme.Opens.instIsOverι, Scheme.Hom.normalizationCoprodIso_inv_coprodDesc_fromNormalization_assoc, Scheme.IdealSheafData.comapIso_hom_fst, AffineSpace.homOverEquiv_apply, pullbackRestrictIsoRestrict_inv_fst_assoc, Scheme.GlueData.ι_isoLocallyRingedSpace_inv, Scheme.IdealSheafData.map_id, Scheme.iso_inv_base_hom_base_apply, Scheme.Opens.fromSpecStalkOfMem_toSpecΓ_assoc, Scheme.Hom.asFiberHom_fiberToSpecResidueField, Scheme.Modules.pseudofunctor_mapComp_inv_τl, locallyOfFiniteType_comp, Scheme.Pullback.Triplet.exists_preimage, Scheme.comp_appTop, Scheme.Hom.stalkMap_hom_inv_apply, IsZariskiLocalAtTarget.iff_of_openCover, instIsLocallyNoetherianXScheme, sigmaOpenCover_f, topologically_isStableUnderComposition, AffineSpace.over_over, Scheme.stalkMap_inv_hom, Scheme.instIsLocalAtTargetIsomorphismsZariskiPrecoverage, Etale.etale_comp, Scheme.OpenCover.pullbackCoverAffineRefinementObjIso_inv_map_assoc, isLocallyArtinian_iff_openCover, Scheme.comp_base_apply, SpecMap_residueFieldIsoBase_inv, ofArrows_ι_mem_zariskiTopology_of_isColimit, Scheme.AffineOpenCover.openCover_I₀, Scheme.PartialMap.isOver_iff, AffineSpace.toSpecMvPolyIntEquiv_comp, Scheme.Pullback.t'_fst_snd_assoc, SpecMap_residueFieldIsoBase_inv_assoc, IsAffineOpen.isoSpec_hom_base_apply, Scheme.Hom.toNormalization_normalizationPullback_fst_assoc, Scheme.exists_cover_of_mem_grothendieckTopology, AffineSpace.homOfVector_toSpecMvPoly, Scheme.OpenCover.restrict_I₀, Scheme.LocalRepresentability.yonedaGluedToSheaf_app_comp, QuasiSeparated.quasiCompact_diagonal, Proj.basicOpenIsoSpec_hom, Scheme.nonempty_of_isLimit, pullbackSpecIso_hom_fst'_assoc, Scheme.Pullback.openCoverOfLeft_I₀, Scheme.GlueData.glue_condition_assoc, isIso_iff_stalk_iso, Scheme.instIsOverMapStalkMapOverInferInstanceOverClass, Scheme.AffineZariskiSite.cocone_ι_app, geometrically_iff_of_commRing_of_isClosedUnderIsomorphisms, IsAffineOpen.SpecMap_appLE_fromSpec_assoc, Scheme.forgetToLocallyRingedSpace_obj, Scheme.compactSpace_of_isLimit, QuasiCompactCover.homCover, instIsClosedImmersionISchemeOfSubsingletonCarrierCarrierCommRingCat, IsAffineOpen.fromSpec_top, IsSchemeTheoreticallyDominant.pullbackFst, Scheme.Pullback.pullbackFstιToV_snd_assoc, SpecToEquivOfLocalRing_symm_apply, preservesLimitsOfShape_discrete_of_isSheaf_zariskiTopology, exists_mem_of_isClosed_of_nonempty, Scheme.Hom.stalkMap_congr, Scheme.instIsOpenImmersionF, Flat.isIso_of_surjective_of_mono, Scheme.affineOpenCover_f, pullbackSpecIso_inv_fst', Scheme.Pullback.t'_fst_snd, Scheme.Cover.pullbackCoverOver'_f, Scheme.Hom.comp_appTop, Scheme.exists_π_app_comp_eq_of_locallyOfFinitePresentation, IsOpenImmersion.opensRange_pullback_snd_of_left, IsAffineOpen.Spec_map_appLE_fromSpec, IsPreimmersion.instIsMultiplicativeScheme, Scheme.Cover.gluedCoverT'_fst_snd, IsZariskiLocalAtTarget.descendsAlong, exists_map_eq_top, IsPreimmersion.instFstScheme, IsAffineOpen.isLocalization_stalk', Scheme.Cover.RelativeGluingData.instIsLocallyDirectedI₀CompFunctorForgetOfIsThin, pullback_of_geometrically', instHasOfPostcompPropertySchemeIsAffineHomIsSeparated, IsSeparated.instIsMultiplicativeScheme, Scheme.IdealSheafData.glueDataObjHom_comp_assoc, IsImmersion.instLiftSchemeId, UniversallyOpen.instIsStableUnderCompositionScheme, universally_isZariskiLocalAtTarget, Scheme.Hom.appLE_comp_appLE_assoc, Scheme.Hom.liftCoborder_ι, Scheme.IsLocallyDirected.exists_of_pullback_V_V, Scheme.canonicallyOverPullback_over, Scheme.mem_smallGrothendieckTopology, Scheme.Modules.pseudofunctor_left_unitality_assoc, Flat.instIsStableUnderCompositionScheme, instHasAffinePropertyIsomorphismsSchemeAndIsAffineIsIsoCommRingCatAppTop, pullbackRestrictIsoRestrict_hom_morphismRestrict, Scheme.Cover.presieve₀_sigma, Scheme.IdealSheafData.glueDataObjHom_comp, Scheme.GlueData.oneHypercover_I₀, Scheme.Cover.pushforwardIso_X, nonempty_isColimit_binaryCofanMk_of_isCompl, instRespectsIsoSchemeTopologicallyIsClosedMap, Scheme.Hom.inl_toNormalization_normalizationCoprodIso_inv_assoc, Scheme.IdealSheafData.inclusion_id, spread_out_unique_of_isGermInjective, IsAffineOpen.Spec_map_appLE_fromSpec_assoc, instIsIsoSchemeCoprodSpec, IsLocalAtTarget.descendsAlong_inf_quasiCompact, Scheme.AffineZariskiSite.PreservesLocalization.opensRange_map, Scheme.IsLocallyDirected.V_self, Scheme.ProEt.instFullOverForget, Scheme.kerFunctor_obj, Scheme.Pullback.pullbackFstιToV_snd, Scheme.Pullback.t'_fst_fst_fst_assoc, diagonal_Spec_map, instQuasiSeparatedFstScheme, Scheme.Hom.stalkMap_congr_assoc, IsOpenImmersion.opensRange_pullback_fst_of_right, Scheme.Modules.pseudofunctor_mapId_hom_τl, exists_appTop_map_eq_zero_of_isLimit, Scheme.Cover.add_toPreZeroHypercover, IsPreimmersion.instIsStableUnderBaseChangeScheme, universally_isLocalAtSource, Scheme.AffineZariskiSite.cocone_pt, Scheme.LocalRepresentability.glueData_J, isEmpty_pullback_sigmaι_of_ne, instHasOfPostcompPropertySchemeIsOpenImmersionOfIsStableUnderBaseChange, Scheme.stalkMap_inv_hom_assoc, IsOpenImmersion.hasPullback_of_left, Scheme.Spec_obj, Scheme.Cover.RelativeGluingData.preimage_toBase_eq_range_ι, exists_app_map_eq_zero_of_isLimit, IsFinite.instIsStableUnderCompositionScheme, IsIntegralHom.instDescScheme, HasAffineProperty.affineAnd_le_isAffineHom, AffineScheme.forgetToScheme_obj, instIsMultiplicativeSchemeIsAffineHom, ExistsHomHomCompEqCompAux.hb, IsAffineOpen.fromSpec_toSpecΓ_assoc, Scheme.Hom.comp_image, IsOpenImmersion.range_pullbackFst, Scheme.appLE_comp_appLE, Scheme.affineBasisCover_map_range, Scheme.Hom.resLE_comp_resLE, IsLocalAtSource.iff_of_iSup_eq_top, Scheme.Cover.map_prop, ΓSpec_adjunction_homEquiv_eq, Scheme.Cover.LocallyDirected.w, WeaklyEtale.instDiagonalScheme, IsSeparated.instRespectsIsoScheme, Scheme.Modules.pushforwardCongr_inv_app_app, AffineTargetMorphismProperty.respectsIso_of, Scheme.etaleTopology_le_proetaleTopology, SurjectiveOnStalks.stableUnderBaseChange, Scheme.Hom.isoImage_hom_ι, Flat.isStableUnderBaseChange, Flat.instSndScheme, AffineSpace.SpecIso_inv_over, spread_out_unique_of_isGermInjective', Scheme.Hom.instIsLocallyDirectedI₀DirectedCoverCompFunctorNormalizationGlueDataForget, Scheme.Hom.toNormalization_fromNormalization, isAffineHom_π_app, instRespectsIsoSchemeTopologicallyGeneralizingMap, Scheme.Cover.Over.isOver_map, Scheme.iso_hom_base_inv_base_apply, Scheme.toIso_inv_ι, IsOpenImmersion.instIsOpenImmersionMapSchemeLocallyRingedSpaceForgetToLocallyRingedSpace, Scheme.Spec_map, Scheme.Hom.toNormalization_inl_normalizationCoprodIso_hom, Scheme.Hom.opensRange_comp_of_isIso, Scheme.instSubcanonicalProetaleTopology, instRespectsSchemeSmoothIsOpenImmersion, Scheme.Hom.inv_image, isomorphisms_eq_stalkwise, Scheme.homOfLE_ι_assoc, Scheme.IsSeparated.isSeparated_terminal_from, UniversallyInjective.isStableUnderBaseChange, Scheme.Pullback.cocycle_snd_snd, instIsIntegralPullbackSchemeOfGeometricallyIntegralOfFlatOfUniversallyOpenOfIsLocallyNoetherian, Scheme.residueFieldCongr_fromSpecResidueField_assoc, Scheme.Hom.inr_toNormalization_normalizationCoprodIso_inv_assoc, IsIntegralHom.instIsStableUnderBaseChangeScheme, Scheme.instIsClosedImmersionLiftIdOfIsSeparated, IsOpenImmersion.le_monomorphisms, Scheme.Cover.mkOfCovers_f, Scheme.pretopology_monotone, Scheme.Hom.normalizationPullback_snd, Scheme.bot_mem_qcPrecoverage, Scheme.Hom.opensRange_comp, IsFinite.eq_inf, HasAffineProperty.affineAnd_containsIdentities, Scheme.Hom.ι_fromNormalization, Scheme.Hom.comp_base, exists_finite_imageι_comp_morphismRestrict_of_finite_image_preimage, IsIntegralHom.instIsMultiplicativeScheme, Scheme.mem_pretopology_iff, Scheme.Cover.pullbackCoverOverProp'_I₀, affineLocally_le, IsAffineOpen.isCompactOpenCovered, Scheme.IdealSheafData.comap_comp, Scheme.Pullback.t_snd_assoc, instLocallyQuasiFiniteCompScheme, Scheme.LocalRepresentability.glueData_t, GeometricallyIntegral.eq_geometricallyReduced_inf_geometricallyIrreducible, Scheme.residueFieldMap_id, Scheme.GlueData.openCover_I₀, Flat.instRespectsSchemeIsOpenImmersion, Scheme.ProEt.instRepresentablyFlatOverForget, Proj.homOfLE_toBasicOpenOfGlobalSections_ι, Scheme.GlueData.oneHypercover_f, instIsStableUnderBaseChangeSchemeGeometricallyReduced, Scheme.restrictFunctor_obj_left, isLocallyNoetherian_iff_openCover, ExistsHomHomCompEqCompAux.range_g_subset, Scheme.IdealSheafData.subschemeMap_subschemeι_assoc, instGeometricallyIrreducibleFstScheme, AffineSpace.isoOfIsAffine_inv_over, IsOpenImmersion.instPreservesLimitSchemeWalkingCospanCospanForget, FormallyUnramified.instIsStableUnderCompositionScheme, instPreservesColimitOverSchemeTopMorphismPropertyOverForget, instSmoothSndScheme, Scheme.Hom.stalkMap_id, universallyClosed_iff, Scheme.instFullLocallyRingedSpaceForgetToLocallyRingedSpace, Scheme.instIsStableUnderBaseChangeQcPrecoverage, Scheme.zariskiPrecoverage_le_qcPrecoverage, IsPreimmersion.comp_iff, pullbackSpecIso_hom_fst', Scheme.IdealSheafData.glueDataObjι_ι, IsImmersion.instSndScheme, IsOpenImmersion.isoOfRangeEq_hom_fac_assoc, instMonoObjWalkingSpanCompOverSchemeTopMorphismPropertySpanOverForgetForgetForgetNoneWalkingPairSomeMapInitOfIsOpenImmersionLeftDiscretePUnit, Scheme.monObjAsOverPullback_one, Scheme.exists_hom_hom_comp_eq_comp_of_locallyOfFiniteType, Scheme.Pullback.p_comm, instIsOverHomOfLE, Scheme.Modules.pseudofunctor_map_r, isPullback_Spec_map_isPushout, AffineSpace.instSurjectiveOverSchemeInferInstanceOverClass, Scheme.restrictFunctor_map_left, Scheme.Hom.SpecMap_residueFieldMap_fromSpecResidueField, Scheme.comp_app_assoc, Scheme.Pullback.Triplet.ofPoint_y, IsAffineOpen.SpecMap_appLE_fromSpec, AffineSpace.comp_homOfVector_assoc, Scheme.Hom.appLE_comp_appLE, Scheme.GlueData.oneHypercover_Y, Scheme.Pullback.cocycle_snd_fst_snd, isIso_of_isOpenImmersion_of_opensRange_eq_top, Scheme.Cover.ColimitGluingData.functor_map, Scheme.Modules.restrictFunctorId_hom_app_app, Scheme.SpecMap_stalkMap_fromSpecStalk, instIsIsoSchemeMapOfCommRingCat, AffineTargetMorphismProperty.cancel_left_of_respectsIso, IsAffineOpen.isoSpec_hom, IsImmersion.isImmersion_iff_exists_of_quasiCompact, isPullback_opens_inf_le, Scheme.Γ_obj_op, Scheme.Hom.stalkMap_inv_hom_apply, Scheme.Cover.intersectionOfLocallyDirected_f, Scheme.IsLocallyDirected.t_id, quasiCompactCover_iff, Scheme.Pullback.forget_comparison_surjective, HasRingHomProperty.iff_of_source_openCover, Scheme.Hom.isoImage_hom_ι_assoc, Scheme.Pullback.gluing_t', Scheme.zariskiTopology_le_propQCTopology, Scheme.inv_base_hom_base_assoc, Scheme.Hom.stalkMap_hom_inv_assoc, Scheme.Modules.pseudofunctor_mapComp_inv_τr, IsLocalAtSource.sigmaDesc, Scheme.isLocallyDirected_of_equifibered_of_injective, Scheme.GlueData.oneHypercover_p₁, Scheme.PartialMap.compHom_hom, Scheme.Pullback.openCoverOfLeft_f, IsSeparated.comp_iff, spread_out_of_isGermInjective, Scheme.IsLocallyDirected.homOfLE_tAux_assoc, Scheme.Pullback.Triplet.ofPoint_SpecTensorTo, Scheme.OpenCover.restrict_X, UniversallyOpen.instCompScheme, Scheme.JointlySurjective.exists_eq, Scheme.Pullback.hasPullback_of_cover, pullbackSpecIso_hom_snd, Scheme.Hom.comp_appTop_assoc, instSmoothFstScheme, Scheme.Cover.mem_pretopology, Scheme.affineOneHypercover_toPreOneHypercover_toPreZeroHypercover, universallyClosed_respectsIso, instIsReducedPullbackSchemeOfGeometricallyReducedOfFlatOfIsLocallyNoetherian_1, Scheme.propQCPrecoverage_monotone, IsDominant.comp_iff, Scheme.Hom.quasiFiniteAt_comp_iff, instIsDominantCompScheme, Scheme.Pullback.t_fst_fst_assoc, IsFinite.instContainsIdentitiesScheme, Scheme.isEmpty_pullback, IsIntegralHom.instCompScheme, Scheme.coe_homeoOfIso, Scheme.Pullback.pullbackP1Iso_hom_fst, UniversallyOpen.instRespectsIsoScheme, Scheme.openCoverBasicOpenTop_f, pointEquivClosedPoint_symm_apply_coe, Scheme.Cover.pullbackHom_map_assoc, Scheme.Cover.gluedCoverT'_fst_fst_assoc, Scheme.OpenCover.instIsOpenImmersionMapI₀FunctorOfLocallyDirected, Scheme.Cover.ColimitGluingData.pullbackGluedIso_inv_snd, Scheme.Cover.functorOfLocallyDirectedHomBase_app, Scheme.Cover.ColimitGluingData.pullbackGluedIso_inv_fst, IsIntegralHom.eq_universallyClosed_inf_isAffineHom, Scheme.ProEt.mk_right_as, IsPreimmersion.comp, AffineSpace.reindex_id, Scheme.GlueData.ι_jointly_surjective, quasiSeparated_isStableUnderComposition, Scheme.AffineZariskiSite.PreservesLocalization.colimitDesc_preimage, IsZariskiLocalAtSource.comp, Scheme.ProEt.instIsContinuousOverForgetTopologyOverProetaleTopology, IsProper.instRespectsIsoScheme, Scheme.LocalRepresentability.glueData_U, IsFinite.instHasOfPostcompPropertySchemeIsSeparated, instIsZariskiLocalAtTargetDiagonalScheme, Scheme.stalkClosedPointTo_comp, pointsPi_surjective, Scheme.pretopology_eq_inf, Scheme.Opens.toSpecΓ_SpecMap_appLE, IsAffineOpen.fromSpec_toSpecΓ, Scheme.Cover.ColimitGluingData.functor_obj, initial_isEmpty, Scheme.pullbackComparison_forget_surjective, Scheme.residueFieldMap_comp, Scheme.isoSpec_image_zeroLocus, Scheme.bot_mem_propQCTopology, Scheme.IdealSheafData.subschemeMap_subschemeι, Scheme.comp_coeBase, quasiSeparated_isStableUnderBaseChange, Scheme.Spec_stalkClosedPointTo_fromSpecStalk, IsSeparated.instFstScheme, IsAffineOpen.toSpecΓ_isoSpec_inv_assoc, IsOpenImmersion.isoOfRangeEq_inv_fac, Scheme.Pullback.openCoverOfLeftRight_X, HasAffineProperty.diagonal_of_openCover, Scheme.mem_toGrothendieck_smallPretopology, Scheme.IsJointlySurjectivePreserving.exists_preimage_fst_triplet_of_prop, Scheme.Pullback.lift_comp_ι, Scheme.Hom.stalkMap_inv_hom, isSmoothOfRelativeDimension_isStableUnderBaseChange, Scheme.Pullback.gluing_V, Scheme.Hom.inv_app, Scheme.Hom.generate_singleton_mem_propQCTopology, Scheme.Hom.isoOpensRange_inv_comp, Scheme.restrictFunctor_obj_hom, Scheme.Cover.ColimitGluingData.relativeGluingData_natTrans_app, IsOpenImmersion.instPreservesLimitSchemeLocallyRingedSpaceWalkingCospanCospanForgetToLocallyRingedSpace_1, Scheme.IdealSheafData.subschemeFunctor_map, Scheme.Cover.pullbackHom_map, IsOpenImmersion.hasLimit_cospan_forget_of_left', instIsAffinePullbackSchemeOfIsAffineHom, Scheme.Hom.Spec_map_residueFieldMap_fromSpecResidueField, Scheme.instFaithfulLocallyRingedSpaceForgetToLocallyRingedSpace, Scheme.isAffine_affineCover, Scheme.AffineZariskiSite.directedCover_I₀, nonempty_isColimit_Γ_mapCocone, Scheme.exists_hom_isAffine_of_isZariskiLocalAtSource, IsSeparated.instSndScheme, Scheme.exists_isOpenCover_and_isAffine, IsOpenImmersion.instSndScheme, Scheme.Pullback.Triplet.Spec_map_tensorInl_fromSpecResidueField, universallyClosed_eq, coprodMk_inr, Scheme.AffineOpenCover.openCover_X, isSheaf_fpqcTopology_continuousMapPresheaf, Scheme.isoOfEq_hom, QuasiCompactCover.exists_isAffineOpen_of_isCompact, isSheaf_zariskiTopology_continuousMapPresheaf, Scheme.Hom.asFiberHom_fiberToSpecResidueField_assoc, quasiSeparatedSpace_iff_quasiCompact_prod_lift, Scheme.Hom.liftQuotient_comp_assoc, Scheme.Pullback.exists_preimage_pullback, IsAffineOpen.isoSpec_inv_ι, Scheme.subcanonical_zariskiTopology, IsOpenImmersion.mono, Scheme.LocalRepresentability.yonedaGluedToSheaf_app_toGlued, isCommMonObj_of_isProper_of_isIntegral_tensorObj_of_isAlgClosed, IsAffineOpen.isoSpec_hom_appTop, IsImmersion.comp_iff, Scheme.IsLocallyDirected.instHasColimit, instHasFiniteCoproductsOverSchemeTopMorphismProperty, IsOpenImmersion.comp_lift, essImage_Spec, IsOpenImmersion.lift_fac, pullbackSpecIso_hom_fst_assoc, IsOpenImmersion.instFstScheme, ValuativeCriterion.eq, instRespectsIsoSchemeTopologicallyIsOpenMap, instRespectsIsoSchemeTopologicallyIsEmbedding, Scheme.Modules.pseudofunctor_mapId_inv_τl, Scheme.comp_appLE_assoc, spread_out_of_isGermInjective', Scheme.Modules.pseudofunctor_associativity, Scheme.Hom.eqToHom_app, Scheme.proetaleTopology_le_fpqcTopology, Scheme.Cover.sigma_f, Scheme.id_app, IsClosedImmersion.isIso_iff_ker_eq_bot, Scheme.Pullback.Triplet.SpecMap_tensorInl_fromSpecResidueField, IsAffineOpen.toSpecΓ_fromSpec, Scheme.Modules.pseudofunctor_mapId_inv_τr, Scheme.RationalMap.isOver_iff, Scheme.AffineZariskiSite.PreservesLocalization.isOpenImmersion, Scheme.Pullback.affine_hasPullback, Scheme.PartialMap.fromSpecStalkOfMem_toPartialMap, Proj.awayι_comp_map, Scheme.homOfLE_rfl, IsOpenImmersion.lift_app, sourceLocalClosure.instRespectsRightScheme, IsAffineOpen.isoSpec_inv_ι_assoc, UniversallyInjective.universally_injective, Scheme.IdealSheafData.map_ker, IsClosedImmersion.isIso_of_injective_of_isAffine, Scheme.Pullback.openCoverOfLeft_X, Scheme.stalkMap_congr, Spec.homEquiv_symm_apply, descendsAlong_universallyOpen_surjective_inf_flat_inf_quasicompact, Scheme.AffineZariskiSite.restrictIsoSpec_inv_app, instFaithfulOverSchemePullbackOfSurjectiveOfFlatOfLocallyOfFinitePresentation, IsIntegralHom.comp_iff, isClosedImmersion_diagonal_restrict_diagonalCoverDiagonalRange, Scheme.Cover.gluedCover_t', Scheme.Cover.RelativeGluingData.instIsOpenImmersionMapI₀Functor, Scheme.Cover.RelativeGluingData.isPullback_natTrans_ι_toBase, Scheme.Hom.stalkMap_congr_hom, geometrically_iff_of_isClosedUnderIsomorphisms, QuasiCompactCover.instPullback₂Scheme, Scheme.Pullback.diagonalCover_map, Scheme.Pullback.t'_snd_fst_snd_assoc, Scheme.inv_base_hom_base, Scheme.isAffine_of_isLimit, Scheme.ProEt.mk_left, AffineTargetMorphismProperty.diagonal_respectsIso, Scheme.Cover.ι_fromGlued, Scheme.Hom.inl_toNormalization_normalizationCoprodIso_inv, Scheme.Hom.inr_normalizationCoprodIso_hom_fromNormalization, Scheme.Pullback.isAffine_of_isAffine_isAffine_isAffine, Scheme.Hom.presieve₀_cover, Scheme.IdealSheafData.glueDataObjHom_id, sourceLocalClosure.le, pointOfClosedPoint_comp, instHasAffinePropertyDiagonalSchemeDiagonal, Scheme.Cover.mem_grothendieckTopology, Scheme.Spec_stalkClosedPointTo_fromSpecStalk_assoc, SpecMap_ΓSpecIso_inv_toSpecΓ_assoc, descendsAlong_universallyInjective_surjective_inf_flat_inf_quasicompact, pullback_of_geometrically, Scheme.Opens.isoOfLE_inv_ι_assoc, universallyClosed_snd, Scheme.Pullback.gluedLiftPullbackMap_fst, QuasiCompactCover.instSumScheme, exists_etale_isCompl_of_quasiFiniteAt, morphismRestrict_id, instIsSchemeTheoreticallyDominantCompScheme, Spec.preimage_comp, Scheme.isAffine_affineOpenCover, Scheme.Hom.id_image, pullbackSpecIso_hom_snd_assoc
«term_⁻¹ᵁ_» 📖	CompOp	—
«termΓ(_,_)» 📖» "API Documentation")	CompOp	—

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
SpecMap_preimage_basicOpen 📖	mathematical	—	CategoryTheory.Functor.obj Scheme.Opens Spec Preorder.smallCategory TopologicalSpace.Opens TopCat.carrier PresheafedSpace.carrier CommRingCat CommRingCat.instCategory SheafedSpace.toPresheafedSpace LocallyRingedSpace.toSheafedSpace Scheme.toLocallyRingedSpace TopCat.str PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder TopologicalSpace.Opens.map PresheafedSpace.Hom.base LocallyRingedSpace.Hom.toHom Scheme.Hom.toLRSHom' Spec.map PrimeSpectrum.basicOpen CommRingCat.carrier CommRing.toCommSemiring CommRingCat.commRing DFunLike.coe CategoryTheory.ConcreteCategory.hom RingHom Semiring.toNonAssocSemiring CommSemiring.toSemiring RingHom.instFunLike CommRingCat.instConcreteCategoryRingHomCarrier	—	—
Spec_carrier 📖	mathematical	—	TopCat.carrier PresheafedSpace.carrier CommRingCat CommRingCat.instCategory SheafedSpace.toPresheafedSpace LocallyRingedSpace.toSheafedSpace Scheme.toLocallyRingedSpace Spec PrimeSpectrum CommRingCat.carrier CommRing.toCommSemiring CommRingCat.commRing	—	—
Spec_closedPoint 📖	mathematical	—	DFunLike.coe PresheafedSpace.carrier CommRingCat CommRingCat.instCategory SheafedSpace.toPresheafedSpace LocallyRingedSpace.toSheafedSpace Scheme.toLocallyRingedSpace Spec TopCat.carrier CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap TopCat.str ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier PresheafedSpace.Hom.base LocallyRingedSpace.Hom.toHom Scheme.Hom.toLRSHom' Spec.map IsLocalRing.closedPoint CommRingCat.carrier CommRing.toCommSemiring CommRingCat.commRing	—	IsLocalRing.comap_closedPoint
Spec_presheaf 📖	mathematical	—	PresheafedSpace.presheaf CommRingCat CommRingCat.instCategory SheafedSpace.toPresheafedSpace LocallyRingedSpace.toSheafedSpace Scheme.toLocallyRingedSpace Spec CategoryTheory.ObjectProperty.FullSubcategory.obj CategoryTheory.Functor Opposite TopologicalSpace.Opens TopCat.carrier PrimeSpectrum.Top CommRingCat.carrier CommRingCat.commRing TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder CategoryTheory.Functor.category CategoryTheory.Presheaf.IsSheaf Opens.grothendieckTopology Spec.structureSheaf	—	—
Spec_sheaf 📖	mathematical	—	Scheme.sheaf Spec Spec.structureSheaf CommRingCat.carrier CommRingCat.commRing	—	—
Spec_toLocallyRingedSpace 📖	mathematical	—	Scheme.toLocallyRingedSpace Spec Spec.locallyRingedSpaceObj	—	—
Spec_zeroLocus 📖	mathematical	—	Scheme.zeroLocus Spec Top.top Scheme.Opens OrderTop.toTop Preorder.toLE PartialOrder.toPreorder SemilatticeSup.toPartialOrder Lattice.toSemilatticeSup CompleteLattice.toLattice TopologicalSpace.Opens.instCompleteLattice TopCat.carrier PresheafedSpace.carrier CommRingCat CommRingCat.instCategory SheafedSpace.toPresheafedSpace LocallyRingedSpace.toSheafedSpace Scheme.toLocallyRingedSpace SheafedSpace.instTopologicalSpaceCarrierCarrier BoundedOrder.toOrderTop CompleteLattice.toBoundedOrder PrimeSpectrum.zeroLocus CommRingCat.carrier CommRing.toCommSemiring CommRingCat.commRing Set.preimage CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory TopologicalSpace.Opens.instPartialOrder PresheafedSpace.presheaf Opposite.op DFunLike.coe CategoryTheory.ConcreteCategory.hom RingHom Semiring.toNonAssocSemiring CommSemiring.toSemiring RingHom.instFunLike CommRingCat.instConcreteCategoryRingHomCarrier CategoryTheory.Iso.inv Scheme.ΓSpecIso	—	Set.image_preimage_eq CategoryTheory.ConcreteCategory.bijective_of_isIso CategoryTheory.Iso.isIso_inv Spec_zeroLocus_eq_zeroLocus
Spec_zeroLocus_eq_zeroLocus 📖	mathematical	—	Scheme.zeroLocus Spec Top.top Scheme.Opens OrderTop.toTop Preorder.toLE PartialOrder.toPreorder SemilatticeSup.toPartialOrder Lattice.toSemilatticeSup CompleteLattice.toLattice TopologicalSpace.Opens.instCompleteLattice TopCat.carrier PresheafedSpace.carrier CommRingCat CommRingCat.instCategory SheafedSpace.toPresheafedSpace LocallyRingedSpace.toSheafedSpace Scheme.toLocallyRingedSpace SheafedSpace.instTopologicalSpaceCarrierCarrier BoundedOrder.toOrderTop CompleteLattice.toBoundedOrder Set.image CommRingCat.carrier CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory TopologicalSpace.Opens.instPartialOrder PresheafedSpace.presheaf Opposite.op DFunLike.coe CategoryTheory.ConcreteCategory.hom RingHom Semiring.toNonAssocSemiring CommSemiring.toSemiring CommRing.toCommSemiring CommRingCat.commRing RingHom.instFunLike CommRingCat.instConcreteCategoryRingHomCarrier CategoryTheory.Iso.inv Scheme.ΓSpecIso PrimeSpectrum.zeroLocus	—	Set.ext PrimeSpectrum.mem_basicOpen Set.image_congr basicOpen_eq_of_affine' CategoryTheory.Iso.inv_hom_id_apply
basicOpen_eq_of_affine 📖	mathematical	—	Scheme.basicOpen Spec Top.top Scheme.Opens OrderTop.toTop Preorder.toLE PartialOrder.toPreorder SemilatticeSup.toPartialOrder Lattice.toSemilatticeSup CompleteLattice.toLattice TopologicalSpace.Opens.instCompleteLattice TopCat.carrier PresheafedSpace.carrier CommRingCat CommRingCat.instCategory SheafedSpace.toPresheafedSpace LocallyRingedSpace.toSheafedSpace Scheme.toLocallyRingedSpace SheafedSpace.instTopologicalSpaceCarrierCarrier BoundedOrder.toOrderTop CompleteLattice.toBoundedOrder DFunLike.coe CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory TopologicalSpace.Opens.instPartialOrder PresheafedSpace.presheaf Opposite.op CommRingCat.carrier CategoryTheory.ConcreteCategory.hom RingHom Semiring.toNonAssocSemiring CommSemiring.toSemiring CommRing.toCommSemiring CommRingCat.commRing RingHom.instFunLike CommRingCat.instConcreteCategoryRingHomCarrier CategoryTheory.Iso.inv Scheme.ΓSpecIso PrimeSpectrum.basicOpen	—	TopologicalSpace.Opens.ext Set.ext CommRingCat.Colimits.hasColimits_commRingCat PrimeSpectrum.isPrime isUnit_map_iff MonoidWithZeroHomClass.toMonoidHomClass RingHomClass.toMonoidWithZeroHomClass AlgHomClass.toRingHomClass AlgEquivClass.toAlgHomClass AlgEquiv.instAlgEquivClass isLocalHom_equiv RingEquivClass.toMulEquivClass AlgEquivClass.toRingEquivClass AlgEquiv.commutes IsLocalization.AtPrime.isUnit_to_map_iff Localization.isLocalization
basicOpen_eq_of_affine' 📖	mathematical	—	Scheme.basicOpen Spec Top.top Scheme.Opens OrderTop.toTop Preorder.toLE PartialOrder.toPreorder SemilatticeSup.toPartialOrder Lattice.toSemilatticeSup CompleteLattice.toLattice TopologicalSpace.Opens.instCompleteLattice TopCat.carrier PresheafedSpace.carrier CommRingCat CommRingCat.instCategory SheafedSpace.toPresheafedSpace LocallyRingedSpace.toSheafedSpace Scheme.toLocallyRingedSpace SheafedSpace.instTopologicalSpaceCarrierCarrier BoundedOrder.toOrderTop CompleteLattice.toBoundedOrder PrimeSpectrum.basicOpen CommRingCat.carrier CommRing.toCommSemiring CommRingCat.commRing DFunLike.coe CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory TopologicalSpace.Opens.instPartialOrder PresheafedSpace.presheaf Opposite.op CategoryTheory.ConcreteCategory.hom RingHom Semiring.toNonAssocSemiring CommSemiring.toSemiring RingHom.instFunLike CommRingCat.instConcreteCategoryRingHomCarrier CategoryTheory.Iso.hom Scheme.ΓSpecIso	—	CategoryTheory.Iso.hom_inv_id_apply basicOpen_eq_of_affine
germ_eq_zero_of_pow_mul_eq_zero 📖	mathematical	TopCat.carrier PresheafedSpace.carrier CommRingCat CommRingCat.instCategory SheafedSpace.toPresheafedSpace LocallyRingedSpace.toSheafedSpace Scheme.toLocallyRingedSpace Scheme.Opens SetLike.instMembership TopologicalSpace.Opens.instSetLike SheafedSpace.instTopologicalSpaceCarrierCarrier Scheme.basicOpen TopologicalSpace.Opens CommRingCat.carrier CategoryTheory.Functor.obj Opposite TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder PresheafedSpace.presheaf Opposite.op Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing CommRing.toNonUnitalCommRing CommRingCat.commRing Monoid.toPow MonoidWithZero.toMonoid Semiring.toMonoidWithZero CommSemiring.toSemiring CommRing.toCommSemiring MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass	DFunLike.coe CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier PresheafedSpace.carrier CommRingCat CommRingCat.instCategory SheafedSpace.toPresheafedSpace LocallyRingedSpace.toSheafedSpace Scheme.toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder PresheafedSpace.presheaf Opposite.op TopCat.Presheaf.stalk CommRingCat.Colimits.hasColimits_commRingCat SheafedSpace.instTopologicalSpaceCarrierCarrier SetLike.instMembership TopologicalSpace.Opens.instSetLike CommRingCat.carrier CategoryTheory.ConcreteCategory.hom RingHom Semiring.toNonAssocSemiring CommSemiring.toSemiring CommRing.toCommSemiring CommRingCat.commRing RingHom.instFunLike CommRingCat.instConcreteCategoryRingHomCarrier TopCat.Presheaf.germ MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing CommRing.toNonUnitalCommRing	—	CommRingCat.Colimits.hasColimits_commRingCat map_pow MonoidWithZeroHomClass.toMonoidHomClass RingHomClass.toMonoidWithZeroHomClass RingHom.instRingHomClass IsUnit.pow Scheme.mem_basicOpen IsUnit.mul_right_eq_zero map_mul NonUnitalRingHomClass.toMulHomClass RingHomClass.toNonUnitalRingHomClass map_zero MonoidWithZeroHomClass.toZeroHomClass
instIsIsoSchemeMapOfCommRingCat 📖	mathematical	—	CategoryTheory.IsIso Scheme Scheme.instCategory Spec Spec.map	—	—
instNonemptyCarrierCarrierCommRingCatSpecOfNontrivialCarrier 📖	mathematical	—	TopCat.carrier PresheafedSpace.carrier CommRingCat CommRingCat.instCategory SheafedSpace.toPresheafedSpace LocallyRingedSpace.toSheafedSpace Scheme.toLocallyRingedSpace Spec	—	—

AlgebraicGeometry.Scheme

Definitions

Name	Category	Theorems
SpecΓIdentity 📖	CompOp	3 math math: SpecΓIdentity_hom_app, SpecΓIdentity_app, SpecΓIdentity_inv_app
algebra_section_section_basicOpen 📖	CompOp	5 math math: AlgebraicGeometry.IsAffineOpen.isLocalization_basicOpen, AlgebraicGeometry.isLocalization_away_of_isAffine, AlgebraicGeometry.IsAffineOpen.app_basicOpen_eq_away_map, AlgebraicGeometry.isLocalization_basicOpen_of_qcqs, AlgebraicGeometry.IsAffineOpen.appLE_eq_away_map
arrowStalkMapIsoOfEq 📖	CompOp	—
basicOpen 📖	CompOp	85 math math: Opens.ι_image_basicOpen, AlgebraicGeometry.Proj.fromOfGlobalSections_resLE, map_PrimeSpectrum_basicOpen_of_affine, mem_basicOpen', basicOpen_of_isUnit, AlgebraicGeometry.IsAffineOpen.basicOpen, AlgebraicGeometry.IsAffineOpen.self_le_iSup_basicOpen_iff, AffineZariskiSite.generate_presieveOfSections, AlgebraicGeometry.IsAffineOpen.isLocalization_basicOpen, zeroLocus_singleton, preimage_basicOpen_top, AlgebraicGeometry.IsAffineOpen.basicOpen_fromSpec_app, Opens.ι_image_basicOpen_topIso_inv, AlgebraicGeometry.isBasis_basicOpen, basicOpen_le, Opens.toSpecΓ_preimage_basicOpen, zeroLocus_def, AlgebraicGeometry.Γ_restrict_isLocalization, AlgebraicGeometry.iSup_basicOpen_of_span_eq_top, Hom.preimage_basicOpen_top, AlgebraicGeometry.injective_germ_basicOpen, AlgebraicGeometry.IsAffineOpen.self_le_basicOpen_union_iff, basicOpen_pow, evaluation_eq_zero_iff_notMem_basicOpen, basicOpen_add_le, Hom.preimage_basicOpen, basicOpen_res_eq, image_basicOpen, isNilpotent_iff_basicOpen_eq_bot, AlgebraicGeometry.IsAffineOpen.instIsAffineToSchemeBasicOpen, AlgebraicGeometry.IsAffineOpen.basicOpen_basicOpen_is_basicOpen, AlgebraicGeometry.IsAffineOpen.iSup_basicOpen_eq_self_iff, basicOpen_one, AlgebraicGeometry.Proj.homOfLE_toBasicOpenOfGlobalSections_ι_assoc, map_basicOpen, AlgebraicGeometry.isRetrocompact_basicOpen, AlgebraicGeometry.IsAffineOpen.basicOpenSectionsToAffine_isIso, AlgebraicGeometry.IsAffineOpen.exists_basicOpen_le, IdealSheafData.opensRange_glueDataObjMap, AlgebraicGeometry.isLocalization_away_of_isAffine, AlgebraicGeometry.IsAffineOpen.ι_basicOpen_preimage, basicOpen_appLE, AlgebraicGeometry.IsAffineOpen.fromSpec_preimage_basicOpen, AlgebraicGeometry.isCompactOpen_iff_eq_basicOpen_union, AlgebraicGeometry.Proj.fromOfGlobalSections_morphismRestrict, affineBasicOpen_coe, basicOpen_mul, AlgebraicGeometry.basicOpen_eq_of_affine', AlgebraicGeometry.exists_eq_pow_mul_of_isAffineOpen, Opens.mem_basicOpen_toScheme, AlgebraicGeometry.AffineTargetMorphismProperty.IsLocal.to_basicOpen, mem_basicOpen_top, Opens.ι_image_basicOpen', basicOpen_res, AlgebraicGeometry.exists_basicOpen_le_appLE_of_appLE_of_isAffine, map_basicOpen_map, basicOpen_zero, AlgebraicGeometry.IsAffineOpen.fromSpec_image_basicOpen, AlgebraicGeometry.IsAffineOpen.basicOpen_union_eq_self_iff, mem_basicOpen'', AlgebraicGeometry.basicOpen_eq_of_affine, AlgebraicGeometry.exists_basicOpen_le_affine_inter, AlgebraicGeometry.basicOpen_eq_bot_iff, AlgebraicGeometry.Proj.homOfLE_toBasicOpenOfGlobalSections_ι, AlgebraicGeometry.IsAffineOpen.fromSpec_preimage_basicOpen', basicOpen_restrict, AlgebraicGeometry.IsAffineOpen.app_basicOpen_eq_away_map, preimage_basicOpen, basicOpen_eq_bot_iff_forall_evaluation_eq_zero, toSpecΓ_preimage_basicOpen, openCoverBasicOpenTop_f, AlgebraicGeometry.isLocalization_basicOpen_of_qcqs, AlgebraicGeometry.exists_eq_pow_mul_of_isCompact_of_isQuasiSeparated, IsQuasiAffine.isBasis_basicOpen, AffineZariskiSite.basicOpen_coe, AlgebraicGeometry.isCompact_and_isOpen_iff_finite_and_eq_biUnion_basicOpen, AlgebraicGeometry.Proj.fromOfGlobalSections_preimage_basicOpen, IdealSheafData.map_ideal_basicOpen, mem_zeroLocus_iff, AlgebraicGeometry.instIsAffineHomιBasicOpen, evaluation_ne_zero_iff_mem_basicOpen, AlgebraicGeometry.isCompact_basicOpen, isNilpotent_iff_basicOpen_eq_bot_of_isCompact, AlgebraicGeometry.IsAffineOpen.appLE_eq_away_map, mem_basicOpen
delabAdjoinNotation 📖	CompOp	—
delabCoeFunAppNotation 📖	CompOp	—
delabCoeFunNotation 📖	CompOp	—
empty 📖	CompOp	2 math math: empty_carrier_carrier, empty_presheaf
forget 📖	CompOp	80 math math: coprodPresheafObjIso_hom_fst_assoc, AlgebraicGeometry.coprodSpec_apply, Hom.toNormalization_inl_normalizationCoprodIso_hom_assoc, AlgebraicGeometry.instIsIsoSchemeCoprodComparisonOppositeCommRingCatSpec, Hom.inr_toNormalization_normalizationCoprodIso_inv, AlgebraicGeometry.instIsAffineSigmaObjScheme, AlgebraicGeometry.sigmaOpenCover_X, AlgebraicGeometry.ι_right_coprodIsoSigma_inv, AlgebraicGeometry.ι_sigmaSpec, AlgebraicGeometry.coprodSpec_inr, AlgebraicGeometry.instIsIsoSchemeSigmaSpecOfFinite, Hom.toNormalization_inr_normalizationCoprodIso_hom, forget_map, AlgebraicGeometry.sigmaMk_mk, AlgebraicGeometry.Flat.instDescScheme, coprodPresheafObjIso_hom_snd_assoc, forget_obj, forgetToTop_comp_forget, AlgebraicGeometry.coprodMk_inl, AlgebraicGeometry.instIsAffineHomDescScheme, AlgebraicGeometry.isPullback_inl_inl_coprodMap, AlgebraicGeometry.instIsAffineCoprodScheme, AlgebraicGeometry.Surjective.sigmaDesc_of_union_range_eq_univ, Cover.instIsLocallyDirectedI₀CompFunctorOfLocallyDirectedForget, AlgebraicGeometry.IsZariskiLocalAtSource.sigmaDesc, AlgebraicGeometry.isIso_stalkMap_coprodSpec, GlueData.instPreservesColimitWalkingMultispanProdJMultispanDiagramForget, AlgebraicGeometry.isCompl_range_inl_inr, AlgebraicGeometry.instIsOpenImmersionInrScheme, AlgebraicGeometry.disjoint_opensRange_sigmaι, AlgebraicGeometry.ι_left_coprodIsoSigma_inv, AlgebraicGeometry.IsZariskiLocalAtTarget.coprodMap, AlgebraicGeometry.IsFinite.instDescScheme, Hom.inl_normalizationCoprodIso_hom_fromNormalization_assoc, AlgebraicGeometry.HasAffineProperty.coprodDesc_affineAnd, Hom.inl_normalizationCoprodIso_hom_fromNormalization, AlgebraicGeometry.instIsOpenImmersionMapScheme, AlgebraicGeometry.sigmaι_eq_iff, AlgebraicGeometry.instSurjectiveDescI₀SchemeF, Hom.inr_normalizationCoprodIso_hom_fromNormalization_assoc, coprodPresheafObjIso_hom_snd, AlgebraicGeometry.coprodSpec_coprodMk, AlgebraicGeometry.instMonoObjWalkingSpanCompSchemeSpanForgetNoneWalkingPairSomeMapInitOfIsOpenImmersion, Hom.normalizationCoprodIso_inv_coprodDesc_fromNormalization, AlgebraicGeometry.instIsOpenImmersionInlScheme, AlgebraicGeometry.isOpenImmersion_sigmaDesc, AlgebraicGeometry.coprodSpec_inr_assoc, IsLocallyDirected.instIsLocallyDirectedWidePushoutShapeCompForgetOfIsOpenImmersionMap, coprodPresheafObjIso_hom_fst, AlgebraicGeometry.isCompl_opensRange_inl_inr, AlgebraicGeometry.IsOpenImmersion.instPreservesLimitSchemeWalkingCospanCospanForget_1, AlgebraicGeometry.instIsOpenImmersionSigmaSpec, Hom.toNormalization_inr_normalizationCoprodIso_hom_assoc, AffineZariskiSite.PreservesLocalization.isLocallyDirected, AlgebraicGeometry.coprodSpec_inl_assoc, AlgebraicGeometry.sigmaOpenCover_I₀, AlgebraicGeometry.ι_sigmaSpec_assoc, AlgebraicGeometry.isPullback_inr_inr_coprodMap, AlgebraicGeometry.coprodSpec_inl, AlgebraicGeometry.instIsLocallyDirectedCompSchemeOverOverTopMorphismPropertyForgetForgetForget, Hom.normalizationCoprodIso_inv_coprodDesc_fromNormalization_assoc, AlgebraicGeometry.sigmaOpenCover_f, Cover.RelativeGluingData.instIsLocallyDirectedI₀CompFunctorForgetOfIsThin, Cover.presieve₀_sigma, Hom.inl_toNormalization_normalizationCoprodIso_inv_assoc, AlgebraicGeometry.instIsIsoSchemeCoprodSpec, AlgebraicGeometry.isEmpty_pullback_sigmaι_of_ne, AlgebraicGeometry.IsIntegralHom.instDescScheme, Hom.instIsLocallyDirectedI₀DirectedCoverCompFunctorNormalizationGlueDataForget, Hom.toNormalization_inl_normalizationCoprodIso_hom, Hom.inr_toNormalization_normalizationCoprodIso_inv_assoc, AlgebraicGeometry.IsOpenImmersion.instPreservesLimitSchemeWalkingCospanCospanForget, AlgebraicGeometry.instMonoObjWalkingSpanCompOverSchemeTopMorphismPropertySpanOverForgetForgetForgetNoneWalkingPairSomeMapInitOfIsOpenImmersionLeftDiscretePUnit, Pullback.forget_comparison_surjective, AlgebraicGeometry.IsLocalAtSource.sigmaDesc, isLocallyDirected_of_equifibered_of_injective, pullbackComparison_forget_surjective, AlgebraicGeometry.coprodMk_inr, Hom.inl_toNormalization_normalizationCoprodIso_inv, Hom.inr_normalizationCoprodIso_hom_fromNormalization
forgetToLocallyRingedSpace 📖	CompOp	16 math math: AlgebraicGeometry.IsOpenImmersion.hasLimit_cospan_forget_of_right', restrictFunctorΓ_inv_app, forgetToLocallyRingedSpace_map, AlgebraicGeometry.IsOpenImmersion.hasLimit_cospan_forget_of_left, AlgebraicGeometry.IsOpenImmersion.instPreservesLimitSchemeLocallyRingedSpaceWalkingCospanCospanForgetToLocallyRingedSpace, Γ_def, AlgebraicGeometry.IsOpenImmersion.hasLimit_cospan_forget_of_right, restrictFunctorΓ_hom_app, IsLocallyDirected.instPreservesColimitLocallyRingedSpaceForgetToLocallyRingedSpace, fullyFaithfulForgetToLocallyRingedSpace_preimage_toLRSHom, forgetToLocallyRingedSpace_obj, AlgebraicGeometry.IsOpenImmersion.instIsOpenImmersionMapSchemeLocallyRingedSpaceForgetToLocallyRingedSpace, instFullLocallyRingedSpaceForgetToLocallyRingedSpace, AlgebraicGeometry.IsOpenImmersion.instPreservesLimitSchemeLocallyRingedSpaceWalkingCospanCospanForgetToLocallyRingedSpace_1, AlgebraicGeometry.IsOpenImmersion.hasLimit_cospan_forget_of_left', instFaithfulLocallyRingedSpaceForgetToLocallyRingedSpace
forgetToTop 📖	CompOp	10 math math: forgetToTop_map, forgetToTop_obj, forgetToTop_comp_forget, AlgebraicGeometry.instPreservesColimitsOfShapeSchemeTopCatDiscreteForgetToTopOfSmall, instIsContinuousTopCatForgetToTopZariskiTopologyGrothendieckTopology, GlueData.instPreservesColimitTopCatWalkingMultispanProdJMultispanDiagramForgetToTop, instEpiTopCatPullbackComparisonForgetToTop, AlgebraicGeometry.IsOpenImmersion.instPreservesLimitSchemeTopCatWalkingCospanCospanForgetToTop_1, AlgebraicGeometry.IsOpenImmersion.instPreservesLimitSchemeTopCatWalkingCospanCospanForgetToTop, AlgebraicGeometry.isomorphisms_eq_stalkwise
forgetToTop_obj_eq_coe 📖	MathDef	—
forget_obj_eq_coe 📖	MathDef	—
fullyFaithfulForgetToLocallyRingedSpace 📖	CompOp	1 math math: fullyFaithfulForgetToLocallyRingedSpace_preimage_toLRSHom
hasCoeToTopCat 📖	CompOp	—
homeoOfIso 📖	CompOp	4 math math: homeoOfIso_symm, coe_homeoOfIso_symm, homeoOfIso_apply, coe_homeoOfIso
instCategory 📖	CompOp	1657 math math: Pullback.gluedLiftPullbackMap_snd, AlgebraicGeometry.continuousMapPresheaf_obj, AlgebraicGeometry.AffineSpace.not_isIntegralHom, Hom.resLE_map_assoc, AlgebraicGeometry.Surjective.instFstScheme, AlgebraicGeometry.AffineSpace.map_Spec_map, mem_propQCPrecoverage_iff_exists_quasiCompactCover, GlueData.oneHypercover_I₁, AlgebraicGeometry.instIsStableUnderBaseChangeSchemeGeometricallyIrreducible, AlgebraicGeometry.WeaklyEtale.weaklyEtale_eq_flat_inf_diagonal_flat, coprodPresheafObjIso_hom_fst_assoc, Hom.toPartialMap_hom, AlgebraicGeometry.descendsAlong_isOpenImmersion_surjective_inf_flat_inf_quasicompact', instEffectiveEpiOfLocallyOfFinitePresentationOfSurjectiveOfFlat, map_PrimeSpectrum_basicOpen_of_affine, bot_mem_grothendieckTopology, Modules.pseudofunctor_map_adj, coverOfIsIso_I₀, AlgebraicGeometry.IsImmersion.isStableUnderBaseChange, AlgebraicGeometry.instLocallyQuasiFiniteSndScheme, isoOfEq_inv, Cover.pullbackCoverOverProp_I₀, AlgebraicGeometry.IsAffineOpen.isoSpec_inv_appTop, AlgebraicGeometry.Spec.map_eq_id, AlgebraicGeometry.AffineSpace.map_toSpecMvPoly_assoc, AffineOpenCover.openCover_f, Cover.RelativeGluingData.cover_X, Modules.pushforwardId_inv_app_app, AlgebraicGeometry.IsClosedImmersion.instIsIsoSchemeToImage, AlgebraicGeometry.AffineSpace.map_reindex, AlgebraicGeometry.IsAffineOpen.map_fromSpec_assoc, Pullback.Triplet.specTensorTo_fst, AlgebraicGeometry.instIsAffineXSchemeCover, Hom.opensRange_pullbackSnd, Cover.pullbackCoverOver_X, AlgebraicGeometry.IsProper.instCompScheme, Cover.gluedCoverT'_snd_snd_assoc, AlgebraicGeometry.instCompactSpaceCarrierCarrierCommRingCatEqualizerSchemeOfQuasiSeparatedSpace, Spec_map_stalkMap_fromSpecStalk, Cover.LocallyDirected.directed, ProEt.instIsContinuousCompOverForgetForgetTopologyProetaleTopology, AlgebraicGeometry.descendsAlong_universallyClosed_surjective_inf_flat_inf_quasicompact, Pullback.gluedLiftPullbackMap_snd_assoc, AlgebraicGeometry.instIsAffineXSchemeFiniteSubcover, PartialMap.ext_iff, AlgebraicGeometry.instHasOfPostcompPropertySchemeUniversallyClosedIsSeparated, AlgebraicGeometry.pointOfClosedPoint_comp_assoc, isEmpty_pullback_iff, AlgebraicGeometry.instIsReducedPullbackSchemeOfGeometricallyReducedOfFlatOfIsLocallyNoetherian, AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv, AlgebraicGeometry.IsSeparated.isStableUnderBaseChange, Hom.quasiFiniteLocus_comp, AlgebraicGeometry.isCommMonObj_of_isProper_of_geometricallyIntegral, AlgebraicGeometry.instIsAffinePullbackSchemeOfIsAffineHom_1, IdealSheafData.glueDataObjHom_ι_assoc, AlgebraicGeometry.instIsLocallyArtinianXScheme, AlgebraicGeometry.QuasiCompactCover.of_finite, AlgebraicGeometry.opensCone_pt, Cover.LocallyDirected.ofIsBasisOpensRange_le_iff, AlgebraicGeometry.Proj.basicOpenIsoSpec_inv_ι_assoc, AlgebraicGeometry.coprodSpec_apply, RingHom.IsStableUnderBaseChange.pullback_fst_appTop, AlgebraicGeometry.instCompactSpaceCarrierCarrierCommRingCatPullbackSchemeOfQuasiCompact, Hom.toNormalization_inl_normalizationCoprodIso_hom_assoc, AffineCover.cover_I₀, Hom.toNormalization_normalizationPullback_fst, AlgebraicGeometry.instLocallyQuasiFiniteCompSchemeιQuasiFiniteLocus, AlgebraicGeometry.IsOpenImmersion.hasLimit_cospan_forget_of_right', AlgebraicGeometry.instIsIsoSchemeCoprodComparisonOppositeCommRingCatSpec, AlgebraicGeometry.Etale.instFstScheme, AlgebraicGeometry.IsAffineOpen.map_fromSpec, AlgebraicGeometry.ΓSpec.isIso_adjunction_counit, germ_stalkClosedPointTo_Spec_fromSpecStalk_assoc, AlgebraicGeometry.instQuasiCompactιSchemeOfQuasiSeparatedSpaceCarrierCarrierCommRingCat, AlgebraicGeometry.injective_isStableUnderComposition, affineBasisCover_is_basis, AlgebraicGeometry.instIsZariskiLocalAtSourceDiagonalSchemeOfHasOfPostcompPropertyOfRespectsRightIsOpenImmersion, Opens.toSpecΓ_naturality, AlgebraicGeometry.instLocallyQuasiFiniteFstScheme, Opens.toSpecΓ_SpecMap_presheaf_map_top, Pullback.openCoverOfLeftRight_f, comp_toLRSHom_assoc, restrictFunctorΓ_inv_app, Hom.inr_toNormalization_normalizationCoprodIso_inv, mem_grothendieckTopology_iff, AlgebraicGeometry.preservesColimit_Γ, AlgebraicGeometry.instPreservesLimitSchemeOppositeCommRingCatRightOpΓOfIsAffineHomMapOfCompactSpaceOfQuasiSeparatedSpaceCarrierCarrierObj, Pullback.Triplet.ofPoint_x, OpenCover.map_glueMorphismsOverOfLocallyDirected_left_assoc, instQuasiCompactCover, overGrothendieckTopology_eq_toGrothendieck_overPretopology, AlgebraicGeometry.universallyClosedTypeComp, AlgebraicGeometry.quasiSeparated_comp, AlgebraicGeometry.HasAffineProperty.affineAnd_le_affineAnd, AlgebraicGeometry.Proj.fromOfGlobalSections_toSpecZero_assoc, Hom.exists_isIso_morphismRestrict_toNormalization, AlgebraicGeometry.AffineSpace.toSpecMvPolyIntEquiv_symm_apply, AlgebraicGeometry.morphismRestrict_ι_assoc, fromSpecStalk_toSpecΓ_assoc, Modules.pushforwardCongr_hom_app_app, AlgebraicGeometry.AffineSpace.SpecIso_hom_appTop, AlgebraicGeometry.sourceLocalClosure.instIsMultiplicativeSchemeOfIsStableUnderBaseChange, AlgebraicGeometry.exists_isAffineOpen_preimage_eq, instQuasiCompactCoverQculift, AlgebraicGeometry.IsSeparated.isClosedImmersion_diagonal, AlgebraicGeometry.instIsAffineSigmaObjScheme, zariskiTopology_le_etaleTopology, AlgebraicGeometry.isIso_SpecMap_iff, zariskiTopology_eq, Cover.copy_f, Cover.iUnion_range, AlgebraicGeometry.IsDominant.respectsIso, AlgebraicGeometry.QuasiCompactCover.instCoverOfIsIso, AlgebraicGeometry.pullbackRestrictIsoRestrict_hom_morphismRestrict_assoc, AlgebraicGeometry.Spec.map_surjective, AlgebraicGeometry.IsProper.instFstScheme, AlgebraicGeometry.weaklyEtale_iff, AlgebraicGeometry.isCompact_iff_exists, AlgebraicGeometry.IsAffineOpen.isCompact_pullback_inf, AlgebraicGeometry.instFinitaryExtensiveScheme, AlgebraicGeometry.pointsPi_surjective_of_isAffine, Pullback.t'_snd_fst_fst_assoc, AlgebraicGeometry.isOpenImmersion_isStableUnderComposition, Cover.glued_cover_cocycle_fst, instIsStableUnderCompositionQcPrecoverage, AlgebraicGeometry.IsSeparated.instIsClosedImmersionMapDescScheme, AlgebraicGeometry.sigmaOpenCover_X, AlgebraicGeometry.GeometricallyIrreducible.comp, Hom.iUnion_support_ker_openCover_map_comp, AlgebraicGeometry.UniversallyOpen.instIsStableUnderBaseChangeScheme, AlgebraicGeometry.diagonal_SpecMap, AlgebraicGeometry.AffineTargetMorphismProperty.IsLocal.respectsIso, inv_app, Pullback.openCoverOfBase_I₀, Hom.id_appTop, AlgebraicGeometry.isAffineHom_diagonal_iff, AlgebraicGeometry.ι_right_coprodIsoSigma_inv, AlgebraicGeometry.ι_sigmaSpec, proetalePrecoverage_le_precoverage_weaklyEtale, AlgebraicGeometry.instHasColimitOverSchemeTopMorphismProperty, Cover.instIsIsoFromGlued, Cover.copy_X, Hom.le_ker_comp, AlgebraicGeometry.Spec.map_injective, AlgebraicGeometry.coprodSpec_inr, AlgebraicGeometry.IsOpenImmersion.comp_lift_assoc, AlgebraicGeometry.IsProper.instSndScheme, OpenCover.finiteSubcover_f, AlgebraicGeometry.instIsIsoSchemeSigmaSpecOfFinite, Hom.singleton_mem_fpqcPrecoverage, Spec_map_stalkSpecializes_fromSpecStalk, Hom.toNormalization_normalizationDesc, Pullback.openCoverOfBase_f, AlgebraicGeometry.HasRingHomProperty.comp_of_isOpenImmersion, ideal_ker_le_ker_ΓSpecIso_inv_comp, Cover.pullbackCoverOver'_X, stalkMap_congr_assoc, Pullback.openCoverOfRight_I₀, Opens.toSpecΓ_SpecMap_map, isoSpec_inv_toSpecΓ, Pullback.SpecTensorTo_SpecOfPoint, AlgebraicGeometry.UniversallyInjective.iff_diagonal, AlgebraicGeometry.instIsStableUnderCompositionSchemeSmooth, hom_inv_apply, AlgebraicGeometry.pointsPi_injective, Hom.instIsIsoToNormalizationOfIsIntegralHom, Cover.ColimitGluingData.pullbackGluedIso_inv_snd_assoc, AlgebraicGeometry.isNoetherian_iff_of_finite_affine_openCover, instHasPullbacksPrecoverageOfHasPullbacks, AlgebraicGeometry.ExistsHomHomCompEqCompAux.exists_eq, ι_toIso_inv, locallyCoverDense_of_le, AlgebraicGeometry.instIsMultiplicativeSchemeQuasiCompact, instIsOverFromSpecStalkOfMem, AlgebraicGeometry.isPullback_opens_inf, Cover.functorOfLocallyDirected_obj, Cover.ColimitGluingData.isPullback, Hom.preimageIso_hom_ι_assoc, isoOfEq_inv_ι, AlgebraicGeometry.isPullback_fiberToSpecResidueField_of_isPullback, openCoverOfIsOpenCover_I₀, AlgebraicGeometry.instIsOpenImmersionLeftSchemeDiscretePUnitMapWalkingSpanOverTopMorphismPropertySpan, instQuasiCompactCoverForgetQc, AlgebraicGeometry.ExistsHomHomCompEqCompAux.h𝒰S, Hom.toNormalization_inr_normalizationCoprodIso_hom, AlgebraicGeometry.UniversallyOpen.instIsMultiplicativeScheme, exists_hom_isAffine_of_isLocalAtSource, AlgebraicGeometry.HasAffineProperty.isStableUnderBaseChange, AlgebraicGeometry.HasAffineProperty.affineAnd_isStableUnderBaseChange, Cover.glued_cover_cocycle, Pullback.instHasPullbacks, IdealSheafData.glueDataObjHom_ι, affineOpenCover_I₀, AlgebraicGeometry.isImmersion_eq_inf, Cover.gluedCover_t, Pullback.t_fst_snd_assoc, SpecΓIdentity_hom_app, AlgebraicGeometry.locallyOfFiniteType_isStableUnderBaseChange, preimage_comp, AlgebraicGeometry.instIsZariskiLocalAtTargetMonomorphismsScheme, Hom.stalkMap_congr_hom_assoc, Cover.RelativeGluingData.toBase_preimage_eq_opensRange_ι, quasiCompactCover_shrink_iff, Cover.RelativeGluingData.cover_I₀, AlgebraicGeometry.sourceLocalClosure.iff_forall_exists, precoverage_mono, Pullback.Triplet.isPullback_SpecMap_tensor, forget_map, AlgebraicGeometry.sigmaMk_mk, Pullback.Triplet.specTensorTo_base_snd, Hom.isOver_iff, PartialMap.fromSpecStalkOfMem_compHom, AlgebraicGeometry.affineAnd_respectsIso, AlgebraicGeometry.descendsAlong_isomorphisms_surjective_inf_flat_inf_quasicompact, isoSpec_inv_preimage_zeroLocus, AlgebraicGeometry.Flat.instDescScheme, Hom.normalizationDesc_comp_assoc, Cover.pullbackCoverOver_I₀, AlgebraicGeometry.instIsMultiplicativeSchemeIsSchemeTheoreticallyDominant, LocalRepresentability.isRepresentable, AlgebraicGeometry.AffineSpace.map_over_assoc, IdealSheafData.inclusion_subschemeι_assoc, zariskiTopology_le_fpqcTopology, AlgebraicGeometry.IsClosedImmersion.comp, Modules.pseudofunctor_map_l, instIsClosedImmersionιOfIsSeparated, Cover.gluedCover_U, mem_jointlySurjectiveTopology_iff_jointlySurjectivePretopology, exists_hom_comp_eq_comp_of_locallyOfFiniteType, AlgebraicGeometry.IsSeparated.instIsClosedImmersionLiftSchemeId, Pullback.tensorCongr_SpecTensorTo, AlgebraicGeometry.IsSeparated.eq_valuativeCriterion, Hom.SpecMap_residueFieldMap_fromSpecResidueField_assoc, AlgebraicGeometry.instGeometricallyReducedSndScheme, AlgebraicGeometry.instCoproductsOfShapeDisjointSchemeOfSmall, OpenCover.map_glueMorphismsOfLocallyDirected_assoc, AlgebraicGeometry.IsImmersion.comp, LocalRepresentability.instIsLocallySurjectiveHomYonedaGluedToSheafOfIsLocallySurjectiveZariskiTopologyDescFunctorOppositeType, Hom.toNormalization_fromNormalization_assoc, AlgebraicGeometry.Flat.epi_of_flat_of_surjective, forgetToTop_map, AlgebraicGeometry.UniversallyOpen.out, instSubcanonicalFppfTopology, AlgebraicGeometry.IsClosedImmersion.eq_proper_inf_monomorphisms, bot_mem_precoverage, Cover.ofQuasiCompactCover_toPreZeroHypercover, AlgebraicGeometry.IsImmersion.instιScheme, AlgebraicGeometry.isSmooth_isStableUnderBaseChange, coprodPresheafObjIso_hom_snd_assoc, AlgebraicGeometry.instGeometricallyIrreducibleCompSchemeOfIsOpenImmersionOfSurjective, AlgebraicGeometry.AffineScheme.mk_obj, isoSpec_hom, ι_toIso_inv_assoc, Hom.id_preimage, Modules.pseudofunctor_right_unitality_assoc, Hom.toImage_imageι, germ_stalkClosedPointTo_Spec_fromSpecStalk, AlgebraicGeometry.SpecToEquivOfLocalRing_apply_snd_coe, OpenCover.pullbackCoverAffineRefinementObjIso_inv_pullbackHom, AlgebraicGeometry.HasAffineProperty.diagonal_iff, forget_obj, eqToHom_c_app, Pullback.gluedLift_p1, exists_affine_mem_range_and_range_subset, AlgebraicGeometry.instHasOfPostcompPropertySchemeIsClosedImmersionIsSeparated, AlgebraicGeometry.AffineSpace.homOverEquiv_symm_apply_coe, forgetToTop_obj, affineOpenCover_idx, Opens.isoOfLE_hom_ι_assoc, Hom.fromNormalization_preimage, AlgebraicGeometry.IsIntegralHom.instSndScheme, toSpecΓ_naturality_assoc, Pullback.Triplet.snd_SpecTensorTo_apply, Modules.pushforwardComp_inv_app_app, AlgebraicGeometry.LocallyQuasiFinite.comp_iff, AlgebraicGeometry.IsSchemeTheoreticallyDominant.pullbackSnd, AlgebraicGeometry.IsFinite.comp_iff, AlgebraicGeometry.HasRingHomProperty.respects_isOpenImmersion, Cover.ColimitGluingData.cocone_ι_transitionMap, OpenCover.map_glueMorphismsOverOfLocallyDirected_left, Pullback.Triplet.specTensorTo_fst_assoc, isoOfEq_hom_ι, AlgebraicGeometry.WeaklyEtale.instFstScheme, AlgebraicGeometry.IsAffineHom.comp_iff, AlgebraicGeometry.AffineScheme.forgetToScheme_preservesLimits, AlgebraicGeometry.HasRingHomProperty.containsIdentities, presieve₀_mem_precoverage_iff, comp_base, AlgebraicGeometry.IsClosedImmersion.lift_fac_assoc, AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv_appTop_coord, Pullback.instNonemptyCarrierCarrierCommRingCatPullbackOfSubsingleton, AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv_over_assoc, AlgebraicGeometry.isIso_iff_isIso_stalkMap, AlgebraicGeometry.IsZariskiLocalAtSource.iff_of_iSup_eq_top, OpenCover.map_glueMorphismsOfLocallyDirected, isoSpec_hom_naturality, homOfLE_ι, toSpecΓ_isoSpec_inv_assoc, Pullback.residueFieldCongr_inv_residueFieldMap_ofPoint, AlgebraicGeometry.IsOpenImmersion.range_pullbackSnd, Hom.resLE_map, Cover.mkOfCovers_I₀, Hom.comp_app, AlgebraicGeometry.IsIntegralHom.instIsStableUnderCompositionScheme, AlgebraicGeometry.IsSeparated.isSeparated_eq_diagonal_isClosedImmersion, Cover.isSheafFor_sigma_iff, IdealSheafData.range_glueDataObjι_ι_eq_support_inter, Pullback.gluing_t, forgetToTop_comp_forget, Pullback.left_affine_comp_pullback_hasPullback, hom_base_inv_base, AlgebraicGeometry.HasAffineProperty.affineAnd_isStableUnderComposition, AlgebraicGeometry.IsLocalAtSource.iff_of_openCover, GlueData.instHasMulticoequalizerDiagram, AlgebraicGeometry.continuousMapPresheaf_map, forgetToLocallyRingedSpace_map, AlgebraicGeometry.IsOpenImmersion.range_pullback_fst_of_right, AlgebraicGeometry.Proj.ι_comp_map, toSpecΓ_naturality, AlgebraicGeometry.WeaklyEtale.instIsStableUnderBaseChangeScheme, AlgebraicGeometry.IsSeparated.diagonal_isClosedImmersion, AlgebraicGeometry.pointEquivClosedPoint_apply_coe, AlgebraicGeometry.instIsStableUnderBaseChangeSchemeLocallyQuasiFinite, Hom.ι_toNormalization, AlgebraicGeometry.geometrically_eq_universally, Cover.ColimitGluingData.pullbackGluedIso_inv_fst_assoc, AlgebraicGeometry.instGeometricallyIntegralSndScheme, Cover.pullbackCoverOverProp_f, AlgebraicGeometry.Spec.faithful, AlgebraicGeometry.Spec.preimage_id, AlgebraicGeometry.Proj.pullbackAwayιIso_hom_SpecMap_awayMap_right_assoc, AlgebraicGeometry.isSheaf_propQCTopology_iff, homeoOfIso_symm, exists_isAffine_of_isLimit, Cover.ColimitGluingData.transitionCocone_pt, comp_coeBase_assoc, AlgebraicGeometry.instCompactSpaceCarrierCarrierCommRingCatPullbackSchemeOfQuasiCompact_1, comp_appLE, AlgebraicGeometry.Surjective.instIsStableUnderBaseChangeScheme, AlgebraicGeometry.IsLocalIso.le_of_isZariskiLocalAtSource, Pullback.cocycle, AlgebraicGeometry.IsFinite.instIsStableUnderBaseChangeScheme, Pullback.pullbackP1Iso_hom_snd, AlgebraicGeometry.IsProper.isStableUnderBaseChange, AlgebraicGeometry.quasiSeparated_eq_diagonal_is_quasiCompact, AlgebraicGeometry.IsOpenImmersion.hasLimit_cospan_forget_of_left, IdealSheafData.inclusion_subschemeι, AlgebraicGeometry.coprodMk_inl, AlgebraicGeometry.IsClosedImmersion.respectsIso, Opens.toSpecΓ_naturality_assoc, GlueData.openCover_f, AlgebraicGeometry.instSmoothOfRelativeDimensionOfNatNatCompScheme, instHasIsosPrecoverageOfContainsIdentitiesOfRespectsIso, AlgebraicGeometry.IsAffineOpen.isoSpec_inv, AlgebraicGeometry.instIsMultiplicativeSchemeSurjective, Hom.id_app, Γ_obj, Opens.fromSpecStalkOfMem_ι_assoc, AlgebraicGeometry.instIsAffineXSchemeAffineCover, AlgebraicGeometry.isLocallyNoetherian_iff_of_affine_openCover, kerAdjunction_unit_app, Hom.singleton_mem_qcPrecoverage, Hom.liftQuotient_comp, Pullback.carrierEquiv_symm_fst, Hom.isPullback_resLE, ProEt.forget_obj, etalePrecoverage_le_proetalePrecoverage, AlgebraicGeometry.instIsClosedImmersionLeftSchemeDiscretePUnitOneOverSpecOf, Cover.ColimitGluingData.transitionMap_id, AlgebraicGeometry.Proj.pullbackAwayιIso_inv_snd_assoc, Hom.asFiberHom_fiberι_assoc, AlgebraicGeometry.instIsClosedUnderIsomorphismsSchemeIrreducibleSpaceCarrierCarrierCommRingCat, AlgebraicGeometry.instIsAffineHomDescScheme, IdealSheafData.comapIso_hom_snd_assoc, Hom.cover_X, directedAffineCover_f, AlgebraicGeometry.IsImmersion.isPullback_toImage_liftCoborder, pullback_map_isOpenImmersion, AlgebraicGeometry.Flat.instFstScheme, AlgebraicGeometry.instIsOverTerminalScheme, Opens.toSpecΓ_SpecMap_presheaf_map_assoc, IdealSheafData.comap_id, Pullback.range_fst_comp, AlgebraicGeometry.instDescendsAlongSchemeMinMorphismPropertySurjectiveFlatLocallyOfFinitePresentationOfQuasiCompactOfIsZariskiLocalAtTarget, AlgebraicGeometry.ΓSpec.adjunction_counit_app', AlgebraicGeometry.quasiCompact_isStableUnderBaseChange, AlgebraicGeometry.Proj.SpecMap_awayMap_awayι_assoc, isoSpec_hom_naturality_assoc, AlgebraicGeometry.pullbackSpecIso_inv_snd, AlgebraicGeometry.IsImmersion.instMapDescScheme, isoSpec_inv_image_zeroLocus, IdealSheafData.range_glueDataObjι_ι, AlgebraicGeometry.instPreservesColimitsOfShapeSchemeTopCatDiscreteForgetToTopOfSmall, AlgebraicGeometry.HasAffineProperty.affineAnd_eq_of_propertyIsLocal, AlgebraicGeometry.quasiCompact_comp, AlgebraicGeometry.SurjectiveOnStalks.mono_of_injective, Pullback.t'_fst_fst_snd_assoc, AlgebraicGeometry.Etale.instSndScheme, AlgebraicGeometry.Etale.instHasOfPostcompPropertySchemeMinMorphismPropertyLocallyOfFiniteTypeFormallyUnramified, AlgebraicGeometry.IsOpenImmersion.isIso, Cover.gluedCover_V, AlgebraicGeometry.Proj.pullbackAwayιIso_inv_fst, AlgebraicGeometry.isIso_morphismRestrict_iff_isIso_app, hom_base_inv_base_assoc, AlgebraicGeometry.Proj.awayι_comp_map_assoc, AlgebraicGeometry.isPullback_inl_inl_coprodMap, AlgebraicGeometry.IsFinite.eq_proper_inf_locallyQuasiFinite, AlgebraicGeometry.SpecMap_ΓSpecIso_inv_toSpecΓ, AlgebraicGeometry.AffineSpace.map_comp, over_def, AlgebraicGeometry.Proj.map_comp, AlgebraicGeometry.HasAffineProperty.iff_of_openCover, Hom.quasiFiniteAt_comp_iff_of_isOpenImmersion, AlgebraicGeometry.IsOpenImmersion.instπWalkingCospanSchemeCospanOne, AlgebraicGeometry.Proj.basicOpenToSpec_SpecMap_awayMap, AlgebraicGeometry.isBasis_preimage_isAffineOpen, Cover.coconeOfLocallyDirected_ι, AlgebraicGeometry.Flat.instIsMultiplicativeScheme, Cover.ι_fromGlued_assoc, AlgebraicGeometry.universallyInjective_iff, AlgebraicGeometry.instHasFiniteLimitsScheme, GlueData.glue_condition, GlueData.f_open, OpenCover.isOpenCover_opensRange, AlgebraicGeometry.pullbackRestrictIsoRestrict_inv_fst, Cover.ulift_X, AlgebraicGeometry.QuasiCompactCover.instCoverOfIsAffineOfFiniteI₀, Cover.RelativeGluingData.ι_toBase, openCoverOfIsOpenCover_f, AlgebraicGeometry.instIsAffineCoprodScheme, Pullback.Triplet.Spec_map_tensor_isPullback, AlgebraicGeometry.IsOpenImmersion.isoOfRangeEq_inv_fac_assoc, id_appTop, Cover.gluedCoverT'_fst_fst, AlgebraicGeometry.pullbackSpecIso_hom_base, Cover.covers, AlgebraicGeometry.IsOpenImmersion.comp, instIsContinuousTopCatForgetToTopZariskiTopologyGrothendieckTopology, Pullback.pullbackFstιToV_fst_assoc, AlgebraicGeometry.smooth_comp, Pullback.gluedLift_p2, AlgebraicGeometry.ValuativeCriterion.Existence.eq, IsLocallyDirected.openCover_f, isCommMonObj_asOver_pullback, AlgebraicGeometry.QuasiCompactCover.exists_hom, GlueData.isOpen_iff, AlgebraicGeometry.Surjective.sigmaDesc_of_union_range_eq_univ, AlgebraicGeometry.QuasiCompactCover.isCompactOpenCovered_of_isCompact, fromSpecStalk_toSpecΓ, AlgebraicGeometry.IsOpenImmersion.instPreservesLimitSchemeLocallyRingedSpaceWalkingCospanCospanForgetToLocallyRingedSpace, AlgebraicGeometry.WeaklyEtale.flat_diagonal, AlgebraicGeometry.instRespectsIsoSchemeTopologicallyIsOpenEmbedding, isMonHom_fst_id_right, AlgebraicGeometry.IsLocalIso.eq_iInf, AlgebraicGeometry.ValuativeCriterion.Existence.stableUnderBaseChange, Spec_map_stalkMap_fromSpecStalk_assoc, PartialMap.equiv_toPartialMap_iff_of_isSeparated, AlgebraicGeometry.Etale.instIsMultiplicativeScheme, AlgebraicGeometry.isInitial_iff_isEmpty, AlgebraicGeometry.AffineSpace.reindex_over, IdealSheafData.subschemeCover_map_subschemeι, Cover.gluedCoverT'_snd_fst, AlgebraicGeometry.AffineSpace.map_over, isAffine_affineBasisCover, Opens.toSpecΓ_SpecMap_presheaf_map_top_assoc, AlgebraicGeometry.ExistsHomHomCompEqCompAux.exists_index, AlgebraicGeometry.topologically_iso_le, Cover.trans_id, AlgebraicGeometry.sourceLocalClosure.property_coverMap_comp, Cover.exists_lift_trans_eq, Cover.exists_of_f_eq_f, Hom.toImage_imageι_assoc, AlgebraicGeometry.instIsStableUnderCompositionSchemeLocallyQuasiFinite, AlgebraicGeometry.QuasiSeparated.diagonalQuasiCompact, GlueData.instPreservesColimitTopCatWalkingMultispanProdJMultispanDiagramForgetToTop, AlgebraicGeometry.ΓSpec.adjunction_counit_app, AlgebraicGeometry.instRespectsIsoSchemeTopologicallyIsClosedEmbedding, AlgebraicGeometry.instGeometricallyReducedFstScheme, Hom.preimage_smoothLocus_eq, coe_homeoOfIso_symm, AlgebraicGeometry.Spec.map_id, AlgebraicGeometry.instIrreducibleSpaceCarrierCarrierCommRingCatPullbackSchemeOfGeometricallyIrreducibleOfUniversallyOpen, stalkMap_hom_inv_assoc, Pullback.t_snd, AlgebraicGeometry.universallyInjective_eq, AlgebraicGeometry.instHasOfPostcompPropertySchemeQuasiSeparatedTopMorphismProperty, AlgebraicGeometry.Proj.basicOpenIsoSpec_inv_ι, id.base, Γ_def, Hom.isoOpensRange_hom_ι_assoc, AlgebraicGeometry.instFaithfulOverSchemePullbackOfSurjectiveOfFlatOfQuasiCompact, AlgebraicGeometry.AffineTargetMorphismProperty.respectsIso_mk, AlgebraicGeometry.Spec.full, AlgebraicGeometry.etale_le_weaklyEtale, Pullback.Triplet.specTensorTo_snd, directedAffineCover_I₀, AlgebraicGeometry.opensDiagram_map, Hom.isoOpensRange_hom_ι, Cover.RelativeGluingData.ι_toBase_assoc, AlgebraicGeometry.instHasOfPrecompPropertySchemeSurjective, AlgebraicGeometry.morphismRestrict_ι, Hom.comp_appLE, Modules.restrictFunctorComp_hom_app_app, Hom.preimageIso_inv_ι, Pullback.gluing_J, SpecMap_stalkSpecializes_fromSpecStalk, descResidueField_fromSpecResidueField, Cover.RelativeGluingData.equifibered, Pullback.gluing_f, Pullback.pullbackP1Iso_hom_fst_assoc, AlgebraicGeometry.Spec.map_preimage_unop, IdealSheafData.map_comp, AlgebraicGeometry.instIsClosedImmersionSndScheme, AlgebraicGeometry.WeaklyEtale.instCompScheme, instEpiTopCatPullbackComparisonForgetToTop, Modules.restrictFunctorComp_inv_app_app, Cover.instIsLocallyDirectedI₀CompFunctorOfLocallyDirectedForget, AlgebraicGeometry.IsOpenImmersion.isPullback_lift_id, AlgebraicGeometry.IsLocalIso.le_of_isLocalAtSource, AlgebraicGeometry.Proj.pullbackAwayιIso_inv_snd, SpecΓIdentity_app, AlgebraicGeometry.Proj.pullbackAwayιIso_inv_fst_assoc, IdealSheafData.inclusion_comp_assoc, homeoOfIso_apply, AlgebraicGeometry.pullbackSpecIso_inv_fst'_assoc, AlgebraicGeometry.affineLocally_respectsIso, LocalRepresentability.comp_toGlued_eq, AlgebraicGeometry.IsZariskiLocalAtSource.sigmaDesc, AlgebraicGeometry.ExistsHomHomCompEqCompAux.h𝒰X, AlgebraicGeometry.isSheaf_type_propQCTopology_iff, AlgebraicGeometry.isIso_stalkMap_coprodSpec, AlgebraicGeometry.Proj.pullbackAwayιIso_hom_awayι, AlgebraicGeometry.instPreservesColimitsOfShapeOppositeCommRingCatSchemeDiscreteSpecOfFinite, IsLocallyDirected.homOfLE_tAux, AlgebraicGeometry.isOpenImmersion_stableUnderBaseChange, Opens.toSpecΓ_SpecMap_map_assoc, AlgebraicGeometry.hasOfPostcompProperty_isOpenImmersion_of_morphismRestrict, AlgebraicGeometry.IsImmersion.instDiagonalScheme, AlgebraicGeometry.exists_appTop_π_eq_of_isAffine_of_isLimit, IdealSheafData.ker_fst_of_isClosedImmersion, GlueData.instPreservesColimitWalkingMultispanProdJMultispanDiagramForget, Pullback.pullbackP1Iso_hom_ι_assoc, AlgebraicGeometry.specializingMap_respectsIso, OpenCover.restrict_f, Pullback.pullbackFstιToV_fst, instIsOverMapResidueFieldMapOverInferInstanceOverClass, AlgebraicGeometry.sourceLocalClosure.instRespectsIsoScheme, AlgebraicGeometry.WeaklyEtale.instRespectsIsoScheme, OpenCover.iSup_opensRange, GlueData.ι_isOpenImmersion, Pullback.Triplet.specTensorTo_snd_assoc, Cover.mkOfCovers_X, AlgebraicGeometry.IsOpenImmersion.instPreservesLimitSchemeTopCatWalkingCospanCospanForgetToTop_1, AlgebraicGeometry.IsOpenImmersion.hasLimit_cospan_forget_of_right, AlgebraicGeometry.universallyClosed_isStableUnderBaseChange, fppfPrecoverage_le_fpqcPrecoverage, Pullback.ofPointTensor_SpecTensorTo_assoc, AlgebraicGeometry.FormallyUnramified.isOpenImmersion_diagonal, AlgebraicGeometry.AffineTargetMorphismProperty.arrow_mk_iso_iff, AlgebraicGeometry.smooth_isStableUnderBaseChange, AlgebraicGeometry.IsFinite.instCompScheme, Pullback.pullbackP1Iso_inv_snd_assoc, kerAdjunction_counit_app, Pullback.t_fst_fst, AlgebraicGeometry.opensDiagramι_app, Pullback.carrierEquiv_symm_snd, AlgebraicGeometry.IsClosedImmersion.comp_iff, AlgebraicGeometry.Proj.pullbackAwayιIso_hom_SpecMap_awayMap_right, AlgebraicGeometry.instIsOpenImmersionCompSchemeιQuasiFiniteLocusToNormalization, zariskiPrecoverage_le_fppfPrecoverage, AlgebraicGeometry.isCompl_range_inl_inr, AlgebraicGeometry.UniversallyClosed.out, exists_cover_of_mem_pretopology, AlgebraicGeometry.universallyClosed_isStableUnderComposition, Cover.ColimitGluingData.fst_gluedCocone_ι, AlgebraicGeometry.ΓSpec.adjunction_unit_app, Opens.toSpecΓ_top, AlgebraicGeometry.universally_isLocalAtTarget, LocalRepresentability.glueData_openCover_map, AlgebraicGeometry.HasRingHomProperty.stableUnderComposition, AlgebraicGeometry.HasRingHomProperty.isMultiplicative, Pullback.range_map, ProEt.instLocallyCoverDenseOverForgetOverProetaleTopology, Pullback.openCoverOfRight_f, Cover.forgetQc_toPreZeroHypercover, Pullback.t'_snd_snd, AlgebraicGeometry.SurjectiveOnStalks.isEmbedding_pullback, AlgebraicGeometry.AffineSpace.SpecIso_inv_appTop_coord, AlgebraicGeometry.IsOpenImmersion.to_iso, AlgebraicGeometry.AffineSpace.functor_obj_map, AlgebraicGeometry.AffineSpace.instLocallyOfFinitePresentationOverSchemeInferInstanceOverClassOfFinite, AlgebraicGeometry.instSurjectiveCompScheme, Hom.iInf_ker_openCover_map_comp, AffineZariskiSite.restrictIsoSpec_hom_app, AlgebraicGeometry.specTargetImageFactorization_comp_assoc, AlgebraicGeometry.instIsIsoSchemeMorphismRestrict, Cover.LocallyDirected.ofIsBasisOpensRange_trans, AlgebraicGeometry.QuasiCompactCover.of_isOpenMap, AlgebraicGeometry.AffineSpace.map_reindex_assoc, AlgebraicGeometry.instGeometricallyIrreducibleSndScheme, AlgebraicGeometry.instIsMultiplicativeSchemeQuasiSeparated, AlgebraicGeometry.IsLocalAtSource.comp, AlgebraicGeometry.instRespectsIsoSchemeSurjective, AlgebraicGeometry.instSubsingletonOverTerminalScheme, AffineCover.cover_f, AlgebraicGeometry.instIsOpenImmersionInrScheme, AlgebraicGeometry.instHasInitialScheme, AlgebraicGeometry.IsLocalIso.instIsStableUnderBaseChangeScheme, AlgebraicGeometry.SurjectiveOnStalks.comp, Cover.gluedCover_J, stalkMap_congr_hom_assoc, Hom.opensRange_pullbackFst, Cover.ColimitGluingData.transitionCocone_ι_app, AlgebraicGeometry.pullbackRestrictIsoRestrict_hom_ι_assoc, AlgebraicGeometry.exists_map_preimage_eq_map_preimage, Cover.sigmaFunctor_obj, AlgebraicGeometry.compactSpace_iff_quasiCompact, mem_overGrothendieckTopology, AlgebraicGeometry.UniversallyOpen.snd, Modules.pseudofunctor_right_unitality, Modules.pseudofunctor_mapComp_hom_τl, AlgebraicGeometry.sourceLocalClosure.instIsStableUnderCompositionSchemeOfIsStableUnderBaseChange, AlgebraicGeometry.IsAffineOpen.isoSpec_inv_toSpecΓ, AlgebraicGeometry.disjoint_opensRange_sigmaι, comp_base_assoc, stalkMap_id, AlgebraicGeometry.isIso_iff_isOpenImmersion_and_surjective, AlgebraicGeometry.instIsAffineHomMapOverSchemeOpensDiagram, AlgebraicGeometry.IsAffineOpen.toSpecΓ_isoSpec_inv, IsLocallyDirected.fst_inv_eq_snd_inv, AlgebraicGeometry.AffineSpace.instIsIsoSchemeOverInferInstanceOverClassOfIsEmpty, Hom.instIsIsoNormalizationPullbackOfSmooth, Opens.toSpecΓ_SpecMap_presheaf_map, AlgebraicGeometry.quasiCompact_isStableUnderComposition, Hom.comp_app_assoc, comp_app, AlgebraicGeometry.instIsStableUnderCompositionSchemeLocallyOfFinitePresentation, AlgebraicGeometry.isIso_iff_isOpenImmersion, AlgebraicGeometry.isOpenImmersion_isMultiplicative, AlgebraicGeometry.AffineSpace.homOfVector_toSpecMvPoly_assoc, AlgebraicGeometry.exists_preimage_eq, Cover.glued_cover_cocycle_snd, Hom.normalizationPullback_snd_assoc, GlueData.ι_eq_iff, Cover.trans_map, AlgebraicGeometry.AffineSpace.comp_homOfVector, openCoverOfIsOpenCover_X, AlgebraicGeometry.ι_left_coprodIsoSigma_inv, AlgebraicGeometry.instHasStrictInitialObjectsScheme, AlgebraicGeometry.AffineSpace.isOpenMap_over, AlgebraicGeometry.instIsLocallyNoetherianPullbackSchemeOfLocallyOfFiniteType_1, isoSpec_inv_naturality_assoc, AlgebraicGeometry.Spec.map_comp_assoc, AlgebraicGeometry.IsImmersion.isImmersion_iff_exists, homOfLE_homOfLE_assoc, AlgebraicGeometry.IsLocalAtSource.respectsLeft_isOpenImmersion, AlgebraicGeometry.instPreservesColimitsOfShapeOppositeCommRingCatSchemeDiscreteWalkingPairSpec, Pullback.pullbackP1Iso_inv_fst, AlgebraicGeometry.Proj.map_id, AlgebraicGeometry.isLocalIso_iff, AlgebraicGeometry.Spec.preimage_comp_assoc, Cover.toSigma_s₀, AlgebraicGeometry.IsClosedImmersion.lift_fac, AlgebraicGeometry.AffineSpace.hom_ext_iff, SpecMap_stalkMap_fromSpecStalk_assoc, OpenCover.pullbackCoverAffineRefinementObjIso_inv_pullbackHom_assoc, AlgebraicGeometry.instHasOfPostcompPropertySchemeQuasiCompactQuasiSeparated, Pullback.pullbackP1Iso_inv_fst_assoc, singleton_mem_precoverage_iff, IdealSheafData.glueDataObjMap_glueDataObjι_assoc, AlgebraicGeometry.QuasiCompactCover.inst, ProEt.topology_eq_inducedTopology, Pullback.tensorCongr_SpecTensorTo_assoc, AlgebraicGeometry.QuasiCompactCover.instPullback₁Scheme, isoSpec_Spec_hom, AlgebraicGeometry.AffineScheme.forgetToScheme_full, IdealSheafData.comapIso_inv_subschemeι_assoc, AlgebraicGeometry.IsZariskiLocalAtTarget.coprodMap, AlgebraicGeometry.IsFinite.instDescScheme, AlgebraicGeometry.AffineSpace.map_toSpecMvPoly, AlgebraicGeometry.Proj.pullbackAwayιIso_hom_SpecMap_awayMap_left, AlgebraicGeometry.instHasFiniteCoproductsScheme, AlgebraicGeometry.AffineSpace.isPullback_map, Hom.inl_normalizationCoprodIso_hom_fromNormalization_assoc, AlgebraicGeometry.exists_appTop_map_eq_zero_of_isAffine_of_isLimit, AlgebraicGeometry.instIsMultiplicativeSchemeUniversallyInjective, AlgebraicGeometry.nonempty_isColimit_cofanMk_of, Cover.pullbackCoverOverProp'_X, AlgebraicGeometry.AffineScheme.forgetToScheme_faithful, AlgebraicGeometry.instQuasiCompactSndScheme, AlgebraicGeometry.pullbackSpecIso_inv_fst, Pullback.t'_snd_fst_fst, AlgebraicGeometry.isIso_of_comp_eq_sigmaSpec, AlgebraicGeometry.IsLocalAtTarget.descendsAlong, AlgebraicGeometry.UniversallyInjective.respectsIso, AlgebraicGeometry.HasAffineProperty.coprodDesc_affineAnd, AlgebraicGeometry.IsFinite.instFstScheme, zariskiPrecoverage_le_propQCPrecoverage, Pullback.affine_affine_hasPullback, grothendieckTopology_eq_inf, AlgebraicGeometry.AffineSpace.homOfVector_over_assoc, AlgebraicGeometry.stalkwise_respectsIso, GlueData.openCover_X, AlgebraicGeometry.morphismRestrict_comp, SpecMap_stalkSpecializes_fromSpecStalk_assoc, Hom.comp_appLE_assoc, isPullback_toSpecΓ_toSpecΓ, Cover.LocallyDirected.trans_comp, Hom.stalkMap_hom_inv, AlgebraicGeometry.instIsMultiplicativeSchemeLocallyOfFiniteType, Cover.pullbackCoverOverProp'_f, restrictFunctorΓ_hom_app, Cover.functorOfLocallyDirected_map, stalkMap_congr_hom, instEffectiveEpiOfQuasiCompactOfSurjectiveOfFlat, Modules.pseudofunctor_mapComp_hom_τr, Γ_map_op, Pullback.range_snd_comp, directedAffineCover_X, Cover.property_trans, AlgebraicGeometry.instIsIsoSchemeAppUnitOppositeCommRingCatAdjunctionOfIsAffine, Modules.restrictFunctorId_inv_app_app, Opens.isoOfLE_hom_ι, AlgebraicGeometry.IsClosedImmersion.isStableUnderBaseChange, Opens.over_def, Opens.fromSpecStalkOfMem_ι, Modules.pushforwardComp_hom_app_app, Cover.ColimitGluingData.fst_gluedCocone_ι_assoc, iso_inv_base_hom_base, Hom.inl_normalizationCoprodIso_hom_fromNormalization, toSpecΓ_isoSpec_inv, Hom.singleton_mem_fppfPrecoverage, AlgebraicGeometry.SpecToEquivOfLocalRing_apply_fst, Hom.comp_apply, Cover.sigma_X, homOfLE_homOfLE, smallGrothendieckTopologyOfLE_eq_toGrothendieck_smallPretopology, ProEt.forget_map, AlgebraicGeometry.Proj.awayι_toSpecZero, AlgebraicGeometry.opensDiagram_obj, Pullback.gluing_U, AlgebraicGeometry.WeaklyEtale.instIsMultiplicativeScheme, ProEt.mk_hom, Pullback.t'_fst_fst_fst, Cover.nonempty_of_nonempty, instSmallPrecoverage, ProEt.instFaithfulOverForget, AlgebraicGeometry.IsProper.stableUnderComposition, Pullback.t_id, toIso_inv_ι_assoc, instFullOppositeIdealSheafDataOverSubschemeFunctor, AlgebraicGeometry.isPullback_SpecMap_of_isPushout, AlgebraicGeometry.instIsOpenImmersionMapScheme, Hom.cover_f, presieve₀_mem_qcPrecoverage_iff, AlgebraicGeometry.sigmaι_eq_iff, AlgebraicGeometry.opensCone_π_app, AlgebraicGeometry.instHasColimitOverScheme, Hom.resLE_id, AlgebraicGeometry.isOpenImmersion_respectsIso, isClosedUnderIsomorphisms_quasiCompactCover, stalkMap_comp, AffineZariskiSite.directedCover_X, AlgebraicGeometry.IsPreimmersion.instMonoScheme, Cover.ulift_f, IdealSheafData.comapIso_inv_subschemeι, AlgebraicGeometry.UniversallyOpen.fst, AlgebraicGeometry.instIsRightAdjointCommRingCatOppositeSchemeΓ, OpenCover.instIsOpenImmersionTrans, IdealSheafData.glueDataObjMap_glueDataObjι, Hom.comp_appIso, AlgebraicGeometry.isIso_of_isEmpty, AlgebraicGeometry.instIsStableUnderBaseChangeSchemeGeometricallyIntegral, AlgebraicGeometry.IsOpenImmersion.range_pullback_to_base_of_left, AlgebraicGeometry.instSurjectiveDescI₀SchemeF, AlgebraicGeometry.over_def, AlgebraicGeometry.IsIntegralHom.instFstScheme, AlgebraicGeometry.Proj.homOfLE_toBasicOpenOfGlobalSections_ι_assoc, Cover.ColimitGluingData.cocone_ι_transitionMap_assoc, AlgebraicGeometry.instLocallyOfFinitePresentationFstScheme, Hom.stalkMap_inv_hom_assoc, AlgebraicGeometry.WeaklyEtale.instHasOfPostcompPropertyScheme, AlgebraicGeometry.isAffineHom_isStableUnderBaseChange, Opens.toSpecΓ_SpecMap_appLE_assoc, IdealSheafData.inclusion_comp, Γ_map, Cover.pullbackCoverOver'_I₀, AlgebraicGeometry.AffineScheme.forgetToScheme_map, Cover.ulift_I₀, AlgebraicGeometry.IsIntegralHom.instIsMultiplicativeScheme_1, LocalRepresentability.yoneda_toGlued_yonedaGluedToSheaf_assoc, AlgebraicGeometry.isSeparated_iff, AlgebraicGeometry.Proj.awayι_toSpecZero_assoc, Hom.inr_normalizationCoprodIso_hom_fromNormalization_assoc, AlgebraicGeometry.HasRingHomProperty.inf, instSmallPropQCPrecoverage, AlgebraicGeometry.IsClosedImmersion.instIsMultiplicativeScheme, AlgebraicGeometry.IsOpenImmersion.lift_fac_assoc, Cover.mem_propQCTopology, AlgebraicGeometry.instIsIntegralPullbackSchemeOfGeometricallyIntegralOfFlatOfUniversallyOpenOfIsLocallyNoetherian_1, AlgebraicGeometry.isSheaf_fpqcTopology_continuousMapPresheafAb, AlgebraicGeometry.ValuativeCommSq.commSq, residueFieldCongr_fromSpecResidueField, AlgebraicGeometry.universallyInjective_eq_diagonal, coprodPresheafObjIso_hom_snd, AlgebraicGeometry.IsOpenImmersion.instHasOfPostcompPropertyScheme, AlgebraicGeometry.coprodSpec_coprodMk, Cover.exists_of_trans_eq_trans, instIsStableUnderBaseChangeJointlySurjectivePrecoverage, AlgebraicGeometry.isAffine_affineScheme, Hom.preimageIso_hom_ι, AlgebraicGeometry.IsClosedImmersion.iff_isFinite_and_mono, Cover.sigmaFunctor_map, Pullback.cocycle_fst_fst_snd, AlgebraicGeometry.UniversallyOpen.instIsStableUnderCompositionSchemeTopologicallyIsOpenMap, isoOfEq_hom_ι_assoc, AlgebraicGeometry.isIso_iff_isOpenImmersion_and_epi_base, AlgebraicGeometry.instRespectsSchemeLocallyQuasiFiniteIsOpenImmersion, isoSpec_inv_naturality, AlgebraicGeometry.IsAffineOpen.toSpecΓ_fromSpec_assoc, Hom.ker_comp_of_isIso, Pullback.openCoverOfRight_X, instSubcanonicalFpqcTopology, AlgebraicGeometry.instMonoObjWalkingSpanCompSchemeSpanForgetNoneWalkingPairSomeMapInitOfIsOpenImmersion, Hom.id_appIso, isoOfEq_inv_ι_assoc, Hom.preimageIso_inv_ι_assoc, AlgebraicGeometry.instQuasiSeparatedSndScheme, Modules.pseudofunctor_mapId_hom_τr, Cover.gluedCoverT'_snd_fst_assoc, Hom.asFiberHom_fiberι, IdealSheafData.subschemeFunctor_obj, AlgebraicGeometry.specTargetImageFactorization_comp, comp_appTop_assoc, AlgebraicGeometry.universallyInjective_isStableUnderComposition, AlgebraicGeometry.instHasColimitsOfShapeDiscreteSchemeOfSmall, AlgebraicGeometry.IsFinite.instSndScheme, Cover.ι_glueMorphisms, AlgebraicGeometry.Proj.pullbackAwayιIso_hom_SpecMap_awayMap_left_assoc, Hom.resLE_comp_resLE_assoc, AlgebraicGeometry.IsZariskiLocalAtSource.respectsLeft_isOpenImmersion, AffineZariskiSite.directedCover_f, AlgebraicGeometry.IsImmersion.instIsMultiplicativeScheme, AlgebraicGeometry.instIsOpenImmersionMapSchemeCompOverOverTopMorphismPropertyForgetForget, AlgebraicGeometry.instIsAffineObjOppositeCommRingCatSchemeSpec, AlgebraicGeometry.AffineSpace.reindex_comp, IsLocallyDirected.ι_eq_ι_iff, AlgebraicGeometry.AffineSpace.map_SpecMap, AlgebraicGeometry.emptyIsInitial_to, AlgebraicGeometry.instLocallyOfFiniteTypeSndScheme, isoSpec_Spec, IsLocallyDirected.openCover_I₀, Hom.normalizationDesc_comp, GlueData.oneHypercover_p₂, AlgebraicGeometry.instGeometricallyIntegralFstScheme, AlgebraicGeometry.isOpenImmersion_eq_inf, Pullback.Triplet.specTensorTo_base_fst, AlgebraicGeometry.compactSpace_iff_exists, LocalRepresentability.glueData_V, AlgebraicGeometry.Surjective.comp_iff, AlgebraicGeometry.instHasCoproductsOfShapeOverSchemeTopMorphismPropertyOfSmall, IsLocallyDirected.instPreservesColimitLocallyRingedSpaceForgetToLocallyRingedSpace, AlgebraicGeometry.Proj.fromOfGlobalSections_toSpecZero, Cover.ι_glueMorphisms_assoc, iso_hom_base_inv_base, AlgebraicGeometry.instIsOpenImmersionLeftSchemeDiscretePUnitιOverTopMorphismProperty, AlgebraicGeometry.AffineSpace.SpecIso_inv_over_assoc, AlgebraicGeometry.sourceLocalClosure.instContainsIdentitiesScheme, Hom.singleton_mem_propQCPrecoverage, AlgebraicGeometry.AffineSpace.instIsAffineHomOverSchemeInferInstanceOverClass, AlgebraicGeometry.sourceLocalClosure.instRespectsLeftSchemeOfIsStableUnderBaseChange, stalkMap_hom_inv, AlgebraicGeometry.IsClosedImmersion.eq_inf, AlgebraicGeometry.sourceAffineLocally_respectsIso, instIsIsoIrreducibleComponentιOfIrreducibleSpaceCarrierCarrierCommRingCat, monObjAsOverPullback_mul, AlgebraicGeometry.isClosedMap_isStableUnderComposition, ProEt.instPreservesFiniteLimitsOverForget, AlgebraicGeometry.HasAffineProperty.instRespectsIsoScheme, AlgebraicGeometry.toSpecΓ_SpecMap_ΓSpecIso_inv_assoc, AlgebraicGeometry.instHasTerminalScheme, Modules.pseudofunctor_obj_obj, AlgebraicGeometry.IsClosedImmersion.isIso_lift, AlgebraicGeometry.targetAffineLocally_affineAnd_eq_affineLocally, Hom.comp_preimage, kerFunctor_map, AlgebraicGeometry.IsZariskiLocalAtSource.iff_of_openCover, stalkMap_hom_inv_apply, AlgebraicGeometry.AffineSpace.reindex_comp_assoc, Hom.iInf_ker_openCover_map_comp_apply, AlgebraicGeometry.Surjective.instSndScheme, AlgebraicGeometry.QuasiCompactCover.instBindScheme, Hom.normalizationCoprodIso_inv_coprodDesc_fromNormalization, AlgebraicGeometry.pullbackSpecIso_inv_snd_assoc, AlgebraicGeometry.AffineSpace.instIsOverHomOfVectorOverSchemeInferInstanceOverClass, Cover.pushforwardIso_f, isoOfEq_rfl, inv_appTop, AlgebraicGeometry.isPullback_Spec_map_pushout, GlueData.sheafValGluedMk_val, AlgebraicGeometry.IsIntegralHom.instHasOfPostcompPropertySchemeIsSeparated, Hom.quasiFiniteAt_iff, AlgebraicGeometry.IsProper.comp_iff, AlgebraicGeometry.instIsOpenImmersionInlScheme, LocalRepresentability.instIsIsoSheafZariskiTopologyTypeYonedaGluedToSheaf, LocalRepresentability.glueData_f, AlgebraicGeometry.universallyClosed_fst, IsLocallyDirected.openCover_X, Spec_map_stalkSpecializes_fromSpecStalk_assoc, AlgebraicGeometry.instIsLocallyNoetherianPullbackSchemeOfLocallyOfFiniteType, Hom.exists_mem_and_isIso_morphismRestrict_toNormalization, AlgebraicGeometry.AffineTargetMorphismProperty.cancel_right_of_respectsIso, Cover.copy_I₀, AlgebraicGeometry.HasAffineProperty.descendsAlong_of_affineAnd, fullyFaithfulForgetToLocallyRingedSpace_preimage_toLRSHom, LocalRepresentability.glueData_t', AlgebraicGeometry.IsOpenImmersion.hasPullback_of_right, Cover.RelativeGluingData.cover_f, instIsOverFstOverInferInstanceOverClassId, AlgebraicGeometry.IsImmersion.instFstScheme, AlgebraicGeometry.AffineSpace.toSpecMvPolyIntEquiv_apply, Cover.pullbackCoverOver_f, descResidueField_stalkClosedPointTo_fromSpecResidueField, Pullback.openCoverOfBase_X, AlgebraicGeometry.isOpenImmersion_sigmaDesc, AlgebraicGeometry.IsOpenImmersion.isoOfRangeEq_hom_fac, AlgebraicGeometry.coprodSpec_inr_assoc, instIsStableUnderSupQcPrecoverage, AlgebraicGeometry.AffineSpace.functor_obj_obj, Spec.map_residueFieldIso_inv_eq_fromSpecResidueField, AlgebraicGeometry.IsClosedImmersion.Spec_iff, Pullback.pullbackP1Iso_hom_snd_assoc, IsLocallyDirected.instIsLocallyDirectedWidePushoutShapeCompForgetOfIsOpenImmersionMap, Cover.toPresieveOver_le_arrows_iff, AlgebraicGeometry.isPreimmersion_eq_inf, Hom.comp_toLRSHom, Pullback.cocycle_fst_fst_fst, AlgebraicGeometry.AffineSpace.map_id, Pullback.openCoverOfBase'_f, AlgebraicGeometry.instIsOpenImmersionMapWalkingSpanSchemeSpan, AlgebraicGeometry.AffineSpace.isIntegralHom_over_iff_isEmpty, AlgebraicGeometry.pullbackSpecIso_inv_fst_assoc, Pullback.ofPointTensor_SpecTensorTo, AlgebraicGeometry.WeaklyEtale.instSndScheme, Hom.id_base, AlgebraicGeometry.AffineSpace.functor_map_app, AlgebraicGeometry.UniversallyOpen.eq, LocalRepresentability.yoneda_toGlued_yonedaGluedToSheaf, AlgebraicGeometry.IsClosedImmersion.isIso_of_ker_eq, AlgebraicGeometry.IsOpenImmersion.range_pullback_snd_of_left, AlgebraicGeometry.instIsMultiplicativeSchemeLocallyQuasiFinite, instIsIsoSubschemeιBotIdealSheafData, Cover.gluedCover_f, AlgebraicGeometry.isProper_eq, AlgebraicGeometry.ΓSpec.adjunction_homEquiv_apply, AlgebraicGeometry.UniversallyOpen.universally_isOpenMap, Pullback.t_fst_snd, PartialMap.ofFromSpecStalk_comp, PartialMap.isOver_iff_eq_restrict, Opens.isoOfLE_inv_ι, Hom.ι_toNormalization_assoc, AlgebraicGeometry.instQuasiCompactLiftSchemeIdOfQuasiSeparatedSpaceCarrierCarrierCommRingCat, Hom.Spec_map_residueFieldMap_fromSpecResidueField_assoc, ProEt.instWeaklyEtaleHomDiscretePUnit, AlgebraicGeometry.Flat.comp, Cover.Hom.sigma_h₀, Cover.overEquiv_generate_toPresieveOver_eq_ofArrows, isoSpec_inv_toSpecΓ_assoc, AlgebraicGeometry.Proj.fromOfGlobalSections_morphismRestrict, Cover.LocallyDirected.trans_id, Hom.isoImage_inv_ι, AlgebraicGeometry.AffineSpace.map_comp_assoc, isSeparated_iff_isClosedImmersion_prod_lift, GlueData.ι_isoCarrier_inv, AlgebraicGeometry.IsFinite.instIsMultiplicativeScheme, IsLocallyDirected.glueDataι_naturality, Cover.toSigma_h₀, AlgebraicGeometry.universallyOpen_iff, IdealSheafData.range_glueDataObjι, Hom.isoImage_inv_ι_assoc, AlgebraicGeometry.IsZariskiLocalAtTarget.descendsAlong_inf_quasiCompact, AlgebraicGeometry.locallyOfFinitePresentation_isStableUnderBaseChange, Modules.pushforwardId_hom_app_app, AlgebraicGeometry.pullbackRestrictIsoRestrict_hom_ι, Pullback.gluing_ι, AlgebraicGeometry.HasRingHomProperty.descendsAlong, Pullback.gluedLiftPullbackMap_fst_assoc, Hom.ker_comp, coprodPresheafObjIso_hom_fst, AlgebraicGeometry.ExistsHomHomCompEqCompAux.ha, AlgebraicGeometry.IsOpenImmersion.range_pullback_to_base_of_right, IdealSheafData.comapIso_hom_snd, AlgebraicGeometry.UniversallyClosed.universally_isClosedMap, AlgebraicGeometry.IsLocalAtTarget.iff_of_openCover, AlgebraicGeometry.isRegularEpi_of_flat_of_surjective_of_isAffine, instHasIsosQcPrecoverage, AlgebraicGeometry.smoothOfRelativeDimension_comp, directedAffineCover_trans, Pullback.t'_snd_fst_snd, AlgebraicGeometry.instIsStableUnderCompositionSchemeLocallyOfFiniteType, IsLocallyDirected.instIsOpenImmersionι, AlgebraicGeometry.instQuasiCompactFstScheme, coverOfIsIso_X, AlgebraicGeometry.isCompl_opensRange_inl_inr, AlgebraicGeometry.universallyClosed_eq_universallySpecializing, Pullback.Triplet.fst_SpecTensorTo_apply, AlgebraicGeometry.ΓSpecIso_inv_ΓSpec_adjunction_homEquiv, AlgebraicGeometry.IsLocalIso.instIsMultiplicativeScheme, AlgebraicGeometry.IsOpenImmersion.instPreservesLimitSchemeWalkingCospanCospanForget_1, LocalRepresentability.instIsLocallyInjectiveHomYonedaGluedToSheaf, Cover.LocallyDirected.property_trans, AlgebraicGeometry.instHasOfPostcompPropertySchemeIsProperIsSeparated, Cover.hasColimit_of_locallyDirected, AlgebraicGeometry.instIsOpenImmersionSigmaSpec, AlgebraicGeometry.topologically_respectsIso, AlgebraicGeometry.ExistsHomHomCompEqCompAux.hab, AlgebraicGeometry.Proj.basicOpenToSpec_SpecMap_awayMap_assoc, AlgebraicGeometry.toSpecΓ_SpecMap_ΓSpecIso_inv, AlgebraicGeometry.HasRingHomProperty.isStableUnderBaseChange, AlgebraicGeometry.exists_map_preimage_le_map_preimage, AlgebraicGeometry.AffineSpace.homOfVector_over, AlgebraicGeometry.IsOpenImmersion.isPullback, Hom.inv_appTop, AlgebraicGeometry.Flat.surjective_descendsAlong_surjective_inf_flat_inf_quasicompact, PartialMap.restrict_hom, AlgebraicGeometry.FormallyUnramified.instIsMultiplicativeScheme, AffineZariskiSite.opensRange_relativeGluingData_map, Cover.ColimitGluingData.prop_trans, Hom.toNormalization_inr_normalizationCoprodIso_hom_assoc, qcCoverFamily_property, AlgebraicGeometry.IsProper.instIsMultiplicativeScheme, AlgebraicGeometry.pullbackSpecIso_hom_fst, AlgebraicGeometry.IsProper.eq_valuativeCriterion, Pullback.base_affine_hasPullback, AlgebraicGeometry.locallyOfFinitePresentation_comp, instIsStableUnderCompositionPrecoverageOfIsStableUnderComposition, AlgebraicGeometry.IsClosedImmersion.eq_isFinite_inf_mono, AlgebraicGeometry.quasiSeparated_iff, AlgebraicGeometry.AffineSpace.reindex_over_assoc, IdealSheafData.comapIso_hom_fst_assoc, IsJointlySurjectivePreserving.exists_preimage_snd_triplet_of_prop, Cover.pullbackCoverOverProp_X, AlgebraicGeometry.QuasiCompactCover.instSumScheme_1, AlgebraicGeometry.IsSeparated.stableUnderComposition, AlgebraicGeometry.instIsAffineTerminalScheme, AlgebraicGeometry.Spec.map_comp, AffineZariskiSite.PreservesLocalization.isLocallyDirected, AlgebraicGeometry.exists_app_map_eq_map_of_isLimit, proetalePrecoverage_le_fpqcPrecoverage, AlgebraicGeometry.instIsStableUnderBaseChangeSchemeGeometrically, Pullback.cocycle_snd_fst_fst, Pullback.pullbackP1Iso_inv_snd, AlgebraicGeometry.isIso_of_isClosedImmersion_of_surjective, isPullback_of_openCover, AlgebraicGeometry.instIsMultiplicativeSchemeUniversallyClosed, Cover.gluedCoverT'_fst_snd_assoc, AlgebraicGeometry.instIsOpenImmersionAppOverSchemeOpensDiagramι, Pullback.range_snd, Hom.stalkMap_comp, SpecΓIdentity_inv_app, Pullback.diagonalCoverDiagonalRange_eq_top_of_injective, Cover.coconeOfLocallyDirected_pt, AlgebraicGeometry.instLocallyOfFinitePresentationSndScheme, AlgebraicGeometry.UniversallyClosed.eq_valuativeCriterion, Cover.exists_eq, AlgebraicGeometry.preservesLimit_rightOp_Γ, ofArrows_mem_precoverage_iff, AlgebraicGeometry.isClosedImmersion_equalizer_ι_left, Cover.ColimitGluingData.trans_app_left, AlgebraicGeometry.instRespectsIsoSchemeTopologicallyInjective, AlgebraicGeometry.IsSeparated.instCompScheme, AlgebraicGeometry.FormallyUnramified.instIsStableUnderBaseChangeScheme, AlgebraicGeometry.Spec.homEquiv_apply, IsLocallyDirected.ι_jointly_surjective, AffineCover.cover_X, Pullback.t'_fst_fst_snd, AlgebraicGeometry.instLocallyOfFiniteTypeFstScheme, jointlySurjectiveTopology_eq_toGrothendieck_jointlySurjectivePretopology, AlgebraicGeometry.AffineSpace.isoOfIsAffine_hom_appTop, AlgebraicGeometry.Etale.etale_isStableUnderBaseChange, AlgebraicGeometry.coprodSpec_inl_assoc, Opens.fromSpecStalkOfMem_toSpecΓ, AlgebraicGeometry.Proj.pullbackAwayιIso_hom_awayι_assoc, instLocallyCoverDenseOverTopMorphismPropertyOverForgetOverGrothendieckTopology, isoSpec_Spec_inv, AlgebraicGeometry.Spec.map_eqToHom, affineBasisCover_obj, Hom.comp_toLRSHom_assoc, Cover.ColimitGluingData.transitionMap_comp, AlgebraicGeometry.HasAffineProperty.diagonal_of_openCover_diagonal, AlgebraicGeometry.sigmaOpenCover_I₀, Cover.Hom.sigma_s₀, AlgebraicGeometry.isomorphisms_eq_isOpenImmersion_inf_surjective, grothendieckTopology_monotone, topIso_hom, AlgebraicGeometry.instPreservesColimitsOfShapeOppositeCommRingCatSchemeDiscretePEmptySpec, Hom.map_resLE, Cover.sigma_I₀, instEtaleF, AlgebraicGeometry.IsPreimmersion.instSndScheme, isSeparated_iff, bot_mem_propQCPrecoverage, inv_hom_apply, AlgebraicGeometry.Proj.SpecMap_awayMap_awayι, AlgebraicGeometry.basicOpenIsoSpecAway_inv_homOfLE_assoc, AlgebraicGeometry.exists_mem_of_isClosed_of_nonempty', AlgebraicGeometry.SurjectiveOnStalks.instIsMultiplicativeScheme, AlgebraicGeometry.IsClosedImmersion.iff_isProper_and_mono, instIsStableUnderBaseChangePrecoverageOfIsJointlySurjectivePreservingOfIsStableUnderBaseChange, Spec.map_residueFieldIso_inv_eq_fromSpecResidueField_assoc, Cover.pushforwardIso_I₀, AlgebraicGeometry.sourceLocalClosure.instIsStableUnderBaseChangeScheme, comp_toLRSHom, AlgebraicGeometry.ι_sigmaSpec_assoc, AlgebraicGeometry.instIrreducibleSpaceCarrierCarrierCommRingCatPullbackSchemeOfGeometricallyIrreducibleOfUniversallyOpen_1, Pullback.cocycle_fst_snd, IdealSheafData.subSchemeCover_map_inclusion_assoc, Cover.ColimitGluingData.gluedCocone_pt, AlgebraicGeometry.isPullback_morphismRestrict, AlgebraicGeometry.instIsAffineXSchemeCoverOfIsIsoIsOpenImmersionId, AlgebraicGeometry.isPullback_inr_inr_coprodMap, OpenCover.finiteSubcover_X, AlgebraicGeometry.instIsAffineHomCompScheme, Pullback.range_fst, AlgebraicGeometry.IsOpenImmersion.instPreservesLimitSchemeTopCatWalkingCospanCospanForgetToTop, AlgebraicGeometry.instIsMultiplicativeSchemeIsDominant, fppfTopology_le_fpqcTopology, AlgebraicGeometry.instHasOfPostcompPropertySchemeIsSeparatedTopMorphismProperty, fppfPrecoverage_eq_inf, AlgebraicGeometry.instRespectsIsoSchemeTargetAffineLocallyOfToProperty, AlgebraicGeometry.AffineSpace.isoOfIsAffine_hom, AlgebraicGeometry.instIsClosedImmersionFstScheme, AlgebraicGeometry.AffineTargetMorphismProperty.isStableUnderBaseChange_of_isStableUnderBaseChangeOnAffine_of_isZariskiLocalAtTarget, Hom.resLE_comp_ι_assoc, AlgebraicGeometry.smoothOfRelativeDimension_isStableUnderBaseChange, AlgebraicGeometry.geometrically_inf, Hom.ι_fromNormalization_assoc, Cover.trans_comp, AlgebraicGeometry.instMonoCoprodScheme, stalkMap_inv_hom_apply, AlgebraicGeometry.basicOpenIsoSpecAway_inv_homOfLE, Hom.map_resLE_assoc, AlgebraicGeometry.IsLocalIso.exists_isOpenImmersion, Hom.comp_base_assoc, propQCPrecoverage_le_precoverage, OpenCover.pullbackCoverAffineRefinementObjIso_inv_map, AlgebraicGeometry.coprodSpec_inl, Pullback.pullbackP1Iso_hom_ι, GlueData.oneHypercover_X, AlgebraicGeometry.IsImmersion.instHasOfPostcompPropertySchemeTopMorphismProperty, IdealSheafData.subSchemeCover_map_inclusion, Modules.conjugateEquiv_pullbackId_hom, AlgebraicGeometry.QuasiCompactCover.singleton, Pullback.range_diagonal_subset_diagonalCoverDiagonalRange, AlgebraicGeometry.ΓSpec.adjunction_homEquiv_symm_apply, Pullback.openCoverOfLeftRight_I₀, Hom.toNormalization_normalizationDesc_assoc, AlgebraicGeometry.IsAffineOpen.isoSpec_hom_apply, Modules.conjugateEquiv_pullbackComp_inv, smallGrothendieckTopology_eq_toGrothendieck_smallPretopology, zariskiPrecoverage_le_fpqcPrecoverage, AlgebraicGeometry.IsFinite.eq_isProper_inf_isAffineHom, AlgebraicGeometry.quasiSeparatedSpace_iff_quasiSeparated, AlgebraicGeometry.isPullback_of_isClosedImmersion, Hom.resLE_comp_ι, AlgebraicGeometry.exists_appTop_π_eq_of_isLimit, Hom.liftCoborder_ι_assoc, AlgebraicGeometry.targetAffineLocally_affineAnd_le, AlgebraicGeometry.IsAffine.affine, Pullback.Triplet.Spec_ofPointTensor_SpecTensorTo, Pullback.t'_snd_snd_assoc, IdealSheafData.inclusion_id_assoc, isIso_subschemeι_iff_eq_bot, AlgebraicGeometry.instIsMultiplicativeSchemeSmoothOfRelativeDimensionOfNatNat, AlgebraicGeometry.Etale.instHasOfPostcompPropertyScheme, Pullback.Triplet.ofPoint_s, AlgebraicGeometry.QuasiCompactCover.isCompactOpenCovered_of_isAffineOpen, coverOfIsIso_f, AlgebraicGeometry.Spec.map_inv, Pullback.instHasPullback, Modules.pseudofunctor_left_unitality, Hom.isoOpensRange_inv_comp_assoc, Modules.pseudofunctor_associativity_assoc, instWeaklyEtaleF, instHasPullbacksIsOpenImmersion, Hom.cover_I₀, exists_isQuasiAffine_of_isLimit, AlgebraicGeometry.pullbackSpecIso_hom_base_assoc, AlgebraicGeometry.IsAffineOpen.isoSpec_inv_toSpecΓ_assoc, precoverage_le_qcPrecoverage_of_isOpenMap, Cover.gluedCoverT'_snd_snd, AlgebraicGeometry.universally_isZariskiLocalAtSource, AlgebraicGeometry.instIsLocallyDirectedCompSchemeOverOverTopMorphismPropertyForgetForgetForget, AlgebraicGeometry.isPullback_SpecMap_pushout, Opens.instIsOverι, Hom.normalizationCoprodIso_inv_coprodDesc_fromNormalization_assoc, IdealSheafData.comapIso_hom_fst, AlgebraicGeometry.AffineSpace.homOverEquiv_apply, AlgebraicGeometry.pullbackRestrictIsoRestrict_inv_fst_assoc, GlueData.ι_isoLocallyRingedSpace_inv, IdealSheafData.map_id, iso_inv_base_hom_base_apply, Opens.fromSpecStalkOfMem_toSpecΓ_assoc, Hom.asFiberHom_fiberToSpecResidueField, Modules.pseudofunctor_mapComp_inv_τl, AlgebraicGeometry.locallyOfFiniteType_comp, Pullback.Triplet.exists_preimage, comp_appTop, Hom.stalkMap_hom_inv_apply, AlgebraicGeometry.IsZariskiLocalAtTarget.iff_of_openCover, AlgebraicGeometry.instIsLocallyNoetherianXScheme, AlgebraicGeometry.sigmaOpenCover_f, AlgebraicGeometry.topologically_isStableUnderComposition, AlgebraicGeometry.AffineSpace.over_over, stalkMap_inv_hom, instIsLocalAtTargetIsomorphismsZariskiPrecoverage, AlgebraicGeometry.Etale.etale_comp, OpenCover.pullbackCoverAffineRefinementObjIso_inv_map_assoc, AlgebraicGeometry.isLocallyArtinian_iff_openCover, comp_base_apply, AlgebraicGeometry.SpecMap_residueFieldIsoBase_inv, AlgebraicGeometry.ofArrows_ι_mem_zariskiTopology_of_isColimit, AffineOpenCover.openCover_I₀, PartialMap.isOver_iff, AlgebraicGeometry.AffineSpace.toSpecMvPolyIntEquiv_comp, Pullback.t'_fst_snd_assoc, AlgebraicGeometry.SpecMap_residueFieldIsoBase_inv_assoc, AlgebraicGeometry.IsAffineOpen.isoSpec_hom_base_apply, Hom.toNormalization_normalizationPullback_fst_assoc, exists_cover_of_mem_grothendieckTopology, AlgebraicGeometry.AffineSpace.homOfVector_toSpecMvPoly, OpenCover.restrict_I₀, LocalRepresentability.yonedaGluedToSheaf_app_comp, AlgebraicGeometry.QuasiSeparated.quasiCompact_diagonal, AlgebraicGeometry.Proj.basicOpenIsoSpec_hom, nonempty_of_isLimit, AlgebraicGeometry.pullbackSpecIso_hom_fst'_assoc, Pullback.openCoverOfLeft_I₀, GlueData.glue_condition_assoc, AlgebraicGeometry.isIso_iff_stalk_iso, instIsOverMapStalkMapOverInferInstanceOverClass, AffineZariskiSite.cocone_ι_app, AlgebraicGeometry.geometrically_iff_of_commRing_of_isClosedUnderIsomorphisms, AlgebraicGeometry.IsAffineOpen.SpecMap_appLE_fromSpec_assoc, forgetToLocallyRingedSpace_obj, compactSpace_of_isLimit, AlgebraicGeometry.QuasiCompactCover.homCover, AlgebraicGeometry.instIsClosedImmersionISchemeOfSubsingletonCarrierCarrierCommRingCat, AlgebraicGeometry.IsAffineOpen.fromSpec_top, AlgebraicGeometry.IsSchemeTheoreticallyDominant.pullbackFst, Pullback.pullbackFstιToV_snd_assoc, AlgebraicGeometry.SpecToEquivOfLocalRing_symm_apply, AlgebraicGeometry.preservesLimitsOfShape_discrete_of_isSheaf_zariskiTopology, AlgebraicGeometry.exists_mem_of_isClosed_of_nonempty, Hom.stalkMap_congr, instIsOpenImmersionF, AlgebraicGeometry.Flat.isIso_of_surjective_of_mono, affineOpenCover_f, AlgebraicGeometry.pullbackSpecIso_inv_fst', Pullback.t'_fst_snd, Cover.pullbackCoverOver'_f, Hom.comp_appTop, exists_π_app_comp_eq_of_locallyOfFinitePresentation, AlgebraicGeometry.IsOpenImmersion.opensRange_pullback_snd_of_left, AlgebraicGeometry.IsAffineOpen.Spec_map_appLE_fromSpec, AlgebraicGeometry.IsPreimmersion.instIsMultiplicativeScheme, Cover.gluedCoverT'_fst_snd, AlgebraicGeometry.IsZariskiLocalAtTarget.descendsAlong, AlgebraicGeometry.exists_map_eq_top, AlgebraicGeometry.IsPreimmersion.instFstScheme, AlgebraicGeometry.IsAffineOpen.isLocalization_stalk', Cover.RelativeGluingData.instIsLocallyDirectedI₀CompFunctorForgetOfIsThin, AlgebraicGeometry.pullback_of_geometrically', AlgebraicGeometry.instHasOfPostcompPropertySchemeIsAffineHomIsSeparated, AlgebraicGeometry.IsSeparated.instIsMultiplicativeScheme, IdealSheafData.glueDataObjHom_comp_assoc, AlgebraicGeometry.IsImmersion.instLiftSchemeId, AlgebraicGeometry.UniversallyOpen.instIsStableUnderCompositionScheme, AlgebraicGeometry.universally_isZariskiLocalAtTarget, Hom.appLE_comp_appLE_assoc, Hom.liftCoborder_ι, IsLocallyDirected.exists_of_pullback_V_V, canonicallyOverPullback_over, mem_smallGrothendieckTopology, Modules.pseudofunctor_left_unitality_assoc, AlgebraicGeometry.Flat.instIsStableUnderCompositionScheme, AlgebraicGeometry.instHasAffinePropertyIsomorphismsSchemeAndIsAffineIsIsoCommRingCatAppTop, AlgebraicGeometry.pullbackRestrictIsoRestrict_hom_morphismRestrict, Cover.presieve₀_sigma, IdealSheafData.glueDataObjHom_comp, GlueData.oneHypercover_I₀, Cover.pushforwardIso_X, AlgebraicGeometry.nonempty_isColimit_binaryCofanMk_of_isCompl, AlgebraicGeometry.instRespectsIsoSchemeTopologicallyIsClosedMap, Hom.inl_toNormalization_normalizationCoprodIso_inv_assoc, IdealSheafData.inclusion_id, AlgebraicGeometry.spread_out_unique_of_isGermInjective, AlgebraicGeometry.IsAffineOpen.Spec_map_appLE_fromSpec_assoc, AlgebraicGeometry.instIsIsoSchemeCoprodSpec, AlgebraicGeometry.IsLocalAtTarget.descendsAlong_inf_quasiCompact, AffineZariskiSite.PreservesLocalization.opensRange_map, IsLocallyDirected.V_self, ProEt.instFullOverForget, kerFunctor_obj, Pullback.pullbackFstιToV_snd, Pullback.t'_fst_fst_fst_assoc, AlgebraicGeometry.diagonal_Spec_map, AlgebraicGeometry.instQuasiSeparatedFstScheme, Hom.stalkMap_congr_assoc, AlgebraicGeometry.IsOpenImmersion.opensRange_pullback_fst_of_right, Modules.pseudofunctor_mapId_hom_τl, AlgebraicGeometry.exists_appTop_map_eq_zero_of_isLimit, Cover.add_toPreZeroHypercover, AlgebraicGeometry.IsPreimmersion.instIsStableUnderBaseChangeScheme, AlgebraicGeometry.universally_isLocalAtSource, AffineZariskiSite.cocone_pt, LocalRepresentability.glueData_J, AlgebraicGeometry.isEmpty_pullback_sigmaι_of_ne, AlgebraicGeometry.instHasOfPostcompPropertySchemeIsOpenImmersionOfIsStableUnderBaseChange, stalkMap_inv_hom_assoc, AlgebraicGeometry.IsOpenImmersion.hasPullback_of_left, Spec_obj, Cover.RelativeGluingData.preimage_toBase_eq_range_ι, AlgebraicGeometry.exists_app_map_eq_zero_of_isLimit, AlgebraicGeometry.IsFinite.instIsStableUnderCompositionScheme, AlgebraicGeometry.IsIntegralHom.instDescScheme, AlgebraicGeometry.HasAffineProperty.affineAnd_le_isAffineHom, AlgebraicGeometry.AffineScheme.forgetToScheme_obj, AlgebraicGeometry.instIsMultiplicativeSchemeIsAffineHom, AlgebraicGeometry.ExistsHomHomCompEqCompAux.hb, AlgebraicGeometry.IsAffineOpen.fromSpec_toSpecΓ_assoc, Hom.comp_image, AlgebraicGeometry.IsOpenImmersion.range_pullbackFst, appLE_comp_appLE, affineBasisCover_map_range, Hom.resLE_comp_resLE, AlgebraicGeometry.IsLocalAtSource.iff_of_iSup_eq_top, Cover.map_prop, AlgebraicGeometry.ΓSpec_adjunction_homEquiv_eq, Cover.LocallyDirected.w, AlgebraicGeometry.WeaklyEtale.instDiagonalScheme, AlgebraicGeometry.IsSeparated.instRespectsIsoScheme, Modules.pushforwardCongr_inv_app_app, AlgebraicGeometry.AffineTargetMorphismProperty.respectsIso_of, etaleTopology_le_proetaleTopology, AlgebraicGeometry.SurjectiveOnStalks.stableUnderBaseChange, Hom.isoImage_hom_ι, AlgebraicGeometry.Flat.isStableUnderBaseChange, AlgebraicGeometry.Flat.instSndScheme, AlgebraicGeometry.AffineSpace.SpecIso_inv_over, AlgebraicGeometry.spread_out_unique_of_isGermInjective', Hom.instIsLocallyDirectedI₀DirectedCoverCompFunctorNormalizationGlueDataForget, Hom.toNormalization_fromNormalization, AlgebraicGeometry.isAffineHom_π_app, AlgebraicGeometry.instRespectsIsoSchemeTopologicallyGeneralizingMap, Cover.Over.isOver_map, iso_hom_base_inv_base_apply, toIso_inv_ι, AlgebraicGeometry.IsOpenImmersion.instIsOpenImmersionMapSchemeLocallyRingedSpaceForgetToLocallyRingedSpace, Spec_map, Hom.toNormalization_inl_normalizationCoprodIso_hom, Hom.opensRange_comp_of_isIso, instSubcanonicalProetaleTopology, AlgebraicGeometry.instRespectsSchemeSmoothIsOpenImmersion, Hom.inv_image, AlgebraicGeometry.isomorphisms_eq_stalkwise, homOfLE_ι_assoc, IsSeparated.isSeparated_terminal_from, AlgebraicGeometry.UniversallyInjective.isStableUnderBaseChange, Pullback.cocycle_snd_snd, AlgebraicGeometry.instIsIntegralPullbackSchemeOfGeometricallyIntegralOfFlatOfUniversallyOpenOfIsLocallyNoetherian, residueFieldCongr_fromSpecResidueField_assoc, Hom.inr_toNormalization_normalizationCoprodIso_inv_assoc, AlgebraicGeometry.IsIntegralHom.instIsStableUnderBaseChangeScheme, instIsClosedImmersionLiftIdOfIsSeparated, AlgebraicGeometry.IsOpenImmersion.le_monomorphisms, Cover.mkOfCovers_f, pretopology_monotone, Hom.normalizationPullback_snd, bot_mem_qcPrecoverage, Hom.opensRange_comp, AlgebraicGeometry.IsFinite.eq_inf, AlgebraicGeometry.HasAffineProperty.affineAnd_containsIdentities, Hom.ι_fromNormalization, Hom.comp_base, AlgebraicGeometry.exists_finite_imageι_comp_morphismRestrict_of_finite_image_preimage, AlgebraicGeometry.IsIntegralHom.instIsMultiplicativeScheme, mem_pretopology_iff, Cover.pullbackCoverOverProp'_I₀, AlgebraicGeometry.affineLocally_le, AlgebraicGeometry.IsAffineOpen.isCompactOpenCovered, IdealSheafData.comap_comp, Pullback.t_snd_assoc, AlgebraicGeometry.instLocallyQuasiFiniteCompScheme, LocalRepresentability.glueData_t, AlgebraicGeometry.GeometricallyIntegral.eq_geometricallyReduced_inf_geometricallyIrreducible, residueFieldMap_id, GlueData.openCover_I₀, AlgebraicGeometry.Flat.instRespectsSchemeIsOpenImmersion, ProEt.instRepresentablyFlatOverForget, AlgebraicGeometry.Proj.homOfLE_toBasicOpenOfGlobalSections_ι, GlueData.oneHypercover_f, AlgebraicGeometry.instIsStableUnderBaseChangeSchemeGeometricallyReduced, restrictFunctor_obj_left, AlgebraicGeometry.isLocallyNoetherian_iff_openCover, AlgebraicGeometry.ExistsHomHomCompEqCompAux.range_g_subset, IdealSheafData.subschemeMap_subschemeι_assoc, AlgebraicGeometry.instGeometricallyIrreducibleFstScheme, AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv_over, AlgebraicGeometry.IsOpenImmersion.instPreservesLimitSchemeWalkingCospanCospanForget, AlgebraicGeometry.FormallyUnramified.instIsStableUnderCompositionScheme, AlgebraicGeometry.instPreservesColimitOverSchemeTopMorphismPropertyOverForget, AlgebraicGeometry.instSmoothSndScheme, Hom.stalkMap_id, AlgebraicGeometry.universallyClosed_iff, instFullLocallyRingedSpaceForgetToLocallyRingedSpace, instIsStableUnderBaseChangeQcPrecoverage, zariskiPrecoverage_le_qcPrecoverage, AlgebraicGeometry.IsPreimmersion.comp_iff, AlgebraicGeometry.pullbackSpecIso_hom_fst', IdealSheafData.glueDataObjι_ι, AlgebraicGeometry.IsImmersion.instSndScheme, AlgebraicGeometry.IsOpenImmersion.isoOfRangeEq_hom_fac_assoc, AlgebraicGeometry.instMonoObjWalkingSpanCompOverSchemeTopMorphismPropertySpanOverForgetForgetForgetNoneWalkingPairSomeMapInitOfIsOpenImmersionLeftDiscretePUnit, monObjAsOverPullback_one, exists_hom_hom_comp_eq_comp_of_locallyOfFiniteType, Pullback.p_comm, AlgebraicGeometry.instIsOverHomOfLE, Modules.pseudofunctor_map_r, AlgebraicGeometry.isPullback_Spec_map_isPushout, AlgebraicGeometry.AffineSpace.instSurjectiveOverSchemeInferInstanceOverClass, restrictFunctor_map_left, Hom.SpecMap_residueFieldMap_fromSpecResidueField, comp_app_assoc, Pullback.Triplet.ofPoint_y, AlgebraicGeometry.IsAffineOpen.SpecMap_appLE_fromSpec, AlgebraicGeometry.AffineSpace.comp_homOfVector_assoc, Hom.appLE_comp_appLE, GlueData.oneHypercover_Y, Pullback.cocycle_snd_fst_snd, AlgebraicGeometry.isIso_of_isOpenImmersion_of_opensRange_eq_top, Cover.ColimitGluingData.functor_map, Modules.restrictFunctorId_hom_app_app, SpecMap_stalkMap_fromSpecStalk, AlgebraicGeometry.instIsIsoSchemeMapOfCommRingCat, AlgebraicGeometry.AffineTargetMorphismProperty.cancel_left_of_respectsIso, AlgebraicGeometry.IsAffineOpen.isoSpec_hom, AlgebraicGeometry.IsImmersion.isImmersion_iff_exists_of_quasiCompact, AlgebraicGeometry.isPullback_opens_inf_le, Γ_obj_op, Hom.stalkMap_inv_hom_apply, Cover.intersectionOfLocallyDirected_f, IsLocallyDirected.t_id, AlgebraicGeometry.quasiCompactCover_iff, Pullback.forget_comparison_surjective, AlgebraicGeometry.HasRingHomProperty.iff_of_source_openCover, Hom.isoImage_hom_ι_assoc, Pullback.gluing_t', zariskiTopology_le_propQCTopology, inv_base_hom_base_assoc, Hom.stalkMap_hom_inv_assoc, Modules.pseudofunctor_mapComp_inv_τr, AlgebraicGeometry.IsLocalAtSource.sigmaDesc, isLocallyDirected_of_equifibered_of_injective, GlueData.oneHypercover_p₁, PartialMap.compHom_hom, Pullback.openCoverOfLeft_f, AlgebraicGeometry.IsSeparated.comp_iff, AlgebraicGeometry.spread_out_of_isGermInjective, IsLocallyDirected.homOfLE_tAux_assoc, Pullback.Triplet.ofPoint_SpecTensorTo, OpenCover.restrict_X, AlgebraicGeometry.UniversallyOpen.instCompScheme, JointlySurjective.exists_eq, Pullback.hasPullback_of_cover, AlgebraicGeometry.pullbackSpecIso_hom_snd, Hom.comp_appTop_assoc, AlgebraicGeometry.instSmoothFstScheme, Cover.mem_pretopology, affineOneHypercover_toPreOneHypercover_toPreZeroHypercover, AlgebraicGeometry.universallyClosed_respectsIso, AlgebraicGeometry.instIsReducedPullbackSchemeOfGeometricallyReducedOfFlatOfIsLocallyNoetherian_1, propQCPrecoverage_monotone, AlgebraicGeometry.IsDominant.comp_iff, Hom.quasiFiniteAt_comp_iff, AlgebraicGeometry.instIsDominantCompScheme, Pullback.t_fst_fst_assoc, AlgebraicGeometry.IsFinite.instContainsIdentitiesScheme, isEmpty_pullback, AlgebraicGeometry.IsIntegralHom.instCompScheme, coe_homeoOfIso, Pullback.pullbackP1Iso_hom_fst, AlgebraicGeometry.UniversallyOpen.instRespectsIsoScheme, openCoverBasicOpenTop_f, AlgebraicGeometry.pointEquivClosedPoint_symm_apply_coe, Cover.pullbackHom_map_assoc, Cover.gluedCoverT'_fst_fst_assoc, OpenCover.instIsOpenImmersionMapI₀FunctorOfLocallyDirected, Cover.ColimitGluingData.pullbackGluedIso_inv_snd, Cover.functorOfLocallyDirectedHomBase_app, Cover.ColimitGluingData.pullbackGluedIso_inv_fst, AlgebraicGeometry.IsIntegralHom.eq_universallyClosed_inf_isAffineHom, ProEt.mk_right_as, AlgebraicGeometry.IsPreimmersion.comp, AlgebraicGeometry.AffineSpace.reindex_id, GlueData.ι_jointly_surjective, AlgebraicGeometry.quasiSeparated_isStableUnderComposition, AffineZariskiSite.PreservesLocalization.colimitDesc_preimage, AlgebraicGeometry.IsZariskiLocalAtSource.comp, ProEt.instIsContinuousOverForgetTopologyOverProetaleTopology, AlgebraicGeometry.IsProper.instRespectsIsoScheme, LocalRepresentability.glueData_U, AlgebraicGeometry.IsFinite.instHasOfPostcompPropertySchemeIsSeparated, AlgebraicGeometry.instIsZariskiLocalAtTargetDiagonalScheme, stalkClosedPointTo_comp, AlgebraicGeometry.pointsPi_surjective, pretopology_eq_inf, Opens.toSpecΓ_SpecMap_appLE, AlgebraicGeometry.IsAffineOpen.fromSpec_toSpecΓ, Cover.ColimitGluingData.functor_obj, AlgebraicGeometry.initial_isEmpty, pullbackComparison_forget_surjective, residueFieldMap_comp, isoSpec_image_zeroLocus, bot_mem_propQCTopology, IdealSheafData.subschemeMap_subschemeι, comp_coeBase, AlgebraicGeometry.quasiSeparated_isStableUnderBaseChange, Spec_stalkClosedPointTo_fromSpecStalk, AlgebraicGeometry.IsSeparated.instFstScheme, AlgebraicGeometry.IsAffineOpen.toSpecΓ_isoSpec_inv_assoc, AlgebraicGeometry.IsOpenImmersion.isoOfRangeEq_inv_fac, Pullback.openCoverOfLeftRight_X, AlgebraicGeometry.HasAffineProperty.diagonal_of_openCover, mem_toGrothendieck_smallPretopology, IsJointlySurjectivePreserving.exists_preimage_fst_triplet_of_prop, Pullback.lift_comp_ι, Hom.stalkMap_inv_hom, AlgebraicGeometry.isSmoothOfRelativeDimension_isStableUnderBaseChange, Pullback.gluing_V, Hom.inv_app, Hom.generate_singleton_mem_propQCTopology, Hom.isoOpensRange_inv_comp, restrictFunctor_obj_hom, Cover.ColimitGluingData.relativeGluingData_natTrans_app, AlgebraicGeometry.IsOpenImmersion.instPreservesLimitSchemeLocallyRingedSpaceWalkingCospanCospanForgetToLocallyRingedSpace_1, IdealSheafData.subschemeFunctor_map, Cover.pullbackHom_map, AlgebraicGeometry.IsOpenImmersion.hasLimit_cospan_forget_of_left', AlgebraicGeometry.instIsAffinePullbackSchemeOfIsAffineHom, Hom.Spec_map_residueFieldMap_fromSpecResidueField, instFaithfulLocallyRingedSpaceForgetToLocallyRingedSpace, isAffine_affineCover, AffineZariskiSite.directedCover_I₀, AlgebraicGeometry.nonempty_isColimit_Γ_mapCocone, exists_hom_isAffine_of_isZariskiLocalAtSource, AlgebraicGeometry.IsSeparated.instSndScheme, exists_isOpenCover_and_isAffine, AlgebraicGeometry.IsOpenImmersion.instSndScheme, Pullback.Triplet.Spec_map_tensorInl_fromSpecResidueField, AlgebraicGeometry.universallyClosed_eq, AlgebraicGeometry.coprodMk_inr, AffineOpenCover.openCover_X, AlgebraicGeometry.isSheaf_fpqcTopology_continuousMapPresheaf, isoOfEq_hom, AlgebraicGeometry.QuasiCompactCover.exists_isAffineOpen_of_isCompact, AlgebraicGeometry.isSheaf_zariskiTopology_continuousMapPresheaf, Hom.asFiberHom_fiberToSpecResidueField_assoc, AlgebraicGeometry.quasiSeparatedSpace_iff_quasiCompact_prod_lift, Hom.liftQuotient_comp_assoc, Pullback.exists_preimage_pullback, AlgebraicGeometry.IsAffineOpen.isoSpec_inv_ι, subcanonical_zariskiTopology, AlgebraicGeometry.IsOpenImmersion.mono, LocalRepresentability.yonedaGluedToSheaf_app_toGlued, AlgebraicGeometry.isCommMonObj_of_isProper_of_isIntegral_tensorObj_of_isAlgClosed, AlgebraicGeometry.IsAffineOpen.isoSpec_hom_appTop, AlgebraicGeometry.IsImmersion.comp_iff, IsLocallyDirected.instHasColimit, AlgebraicGeometry.instHasFiniteCoproductsOverSchemeTopMorphismProperty, AlgebraicGeometry.IsOpenImmersion.comp_lift, AlgebraicGeometry.essImage_Spec, AlgebraicGeometry.IsOpenImmersion.lift_fac, AlgebraicGeometry.pullbackSpecIso_hom_fst_assoc, AlgebraicGeometry.IsOpenImmersion.instFstScheme, AlgebraicGeometry.ValuativeCriterion.eq, AlgebraicGeometry.instRespectsIsoSchemeTopologicallyIsOpenMap, AlgebraicGeometry.instRespectsIsoSchemeTopologicallyIsEmbedding, Modules.pseudofunctor_mapId_inv_τl, comp_appLE_assoc, AlgebraicGeometry.spread_out_of_isGermInjective', Modules.pseudofunctor_associativity, Hom.eqToHom_app, proetaleTopology_le_fpqcTopology, Cover.sigma_f, id_app, AlgebraicGeometry.IsClosedImmersion.isIso_iff_ker_eq_bot, Pullback.Triplet.SpecMap_tensorInl_fromSpecResidueField, AlgebraicGeometry.IsAffineOpen.toSpecΓ_fromSpec, Modules.pseudofunctor_mapId_inv_τr, RationalMap.isOver_iff, AffineZariskiSite.PreservesLocalization.isOpenImmersion, Pullback.affine_hasPullback, PartialMap.fromSpecStalkOfMem_toPartialMap, AlgebraicGeometry.Proj.awayι_comp_map, homOfLE_rfl, AlgebraicGeometry.IsOpenImmersion.lift_app, AlgebraicGeometry.sourceLocalClosure.instRespectsRightScheme, AlgebraicGeometry.IsAffineOpen.isoSpec_inv_ι_assoc, AlgebraicGeometry.UniversallyInjective.universally_injective, IdealSheafData.map_ker, AlgebraicGeometry.IsClosedImmersion.isIso_of_injective_of_isAffine, Pullback.openCoverOfLeft_X, stalkMap_congr, AlgebraicGeometry.Spec.homEquiv_symm_apply, AlgebraicGeometry.descendsAlong_universallyOpen_surjective_inf_flat_inf_quasicompact, AffineZariskiSite.restrictIsoSpec_inv_app, AlgebraicGeometry.instFaithfulOverSchemePullbackOfSurjectiveOfFlatOfLocallyOfFinitePresentation, AlgebraicGeometry.IsIntegralHom.comp_iff, AlgebraicGeometry.isClosedImmersion_diagonal_restrict_diagonalCoverDiagonalRange, Cover.gluedCover_t', Cover.RelativeGluingData.instIsOpenImmersionMapI₀Functor, Cover.RelativeGluingData.isPullback_natTrans_ι_toBase, Hom.stalkMap_congr_hom, AlgebraicGeometry.geometrically_iff_of_isClosedUnderIsomorphisms, AlgebraicGeometry.QuasiCompactCover.instPullback₂Scheme, Pullback.diagonalCover_map, Pullback.t'_snd_fst_snd_assoc, inv_base_hom_base, isAffine_of_isLimit, ProEt.mk_left, AlgebraicGeometry.AffineTargetMorphismProperty.diagonal_respectsIso, Cover.ι_fromGlued, Hom.inl_toNormalization_normalizationCoprodIso_inv, Hom.inr_normalizationCoprodIso_hom_fromNormalization, Pullback.isAffine_of_isAffine_isAffine_isAffine, Hom.presieve₀_cover, IdealSheafData.glueDataObjHom_id, AlgebraicGeometry.sourceLocalClosure.le, AlgebraicGeometry.pointOfClosedPoint_comp, AlgebraicGeometry.instHasAffinePropertyDiagonalSchemeDiagonal, Cover.mem_grothendieckTopology, Spec_stalkClosedPointTo_fromSpecStalk_assoc, AlgebraicGeometry.SpecMap_ΓSpecIso_inv_toSpecΓ_assoc, AlgebraicGeometry.descendsAlong_universallyInjective_surjective_inf_flat_inf_quasicompact, AlgebraicGeometry.pullback_of_geometrically, Opens.isoOfLE_inv_ι_assoc, AlgebraicGeometry.universallyClosed_snd, Pullback.gluedLiftPullbackMap_fst, AlgebraicGeometry.QuasiCompactCover.instSumScheme, AlgebraicGeometry.exists_etale_isCompl_of_quasiFiniteAt, AlgebraicGeometry.morphismRestrict_id, AlgebraicGeometry.instIsSchemeTheoreticallyDominantCompScheme, AlgebraicGeometry.Spec.preimage_comp, isAffine_affineOpenCover, Hom.id_image, AlgebraicGeometry.pullbackSpecIso_hom_snd_assoc
instCoeFunHomForallCarrierCarrierCommRingCat 📖	CompOp	—
instCoeSortType 📖	CompOp	—
instEmptyCollection 📖	CompOp	4 math math: emptyTo_base, AlgebraicGeometry.instIsEmptyCarrierCarrierCommRingCatEmptyCollectionScheme, AlgebraicGeometry.emptyIsInitial_to, emptyTo_c_app
instPreorderCarrierCarrierCommRingCat 📖	CompOp	3 math math: height_of_isClosed, AlgebraicGeometry.AffineSpace.spec_le_iff, le_iff_specializes
instUniqueCarrierCarrierCommRingCatSpecOf 📖	CompOp	1 math math: default_asIdeal
sheaf 📖	CompOp	1 math math: AlgebraicGeometry.Spec_sheaf
toLocallyRingedSpace 📖	CompOp	1414 math math: AlgebraicGeometry.continuousMapPresheaf_obj, toSpecΓ_apply, Hom.resLE_map_assoc, Opens.ι_image_basicOpen, AlgebraicGeometry.Γ_map_morphismRestrict, Hom.smoothLocus_eq_top_iff, Modules.pushforward_obj_presheaf_map, ΓSpecIso_inv_naturality_assoc, AlgebraicGeometry.Proj.awayMap_awayToSection_assoc, AlgebraicGeometry.QuasiCompact.isCompact_preimage_singleton, fromSpecStalk_closedPoint, coprodPresheafObjIso_hom_fst_assoc, AlgebraicGeometry.Proj.fromOfGlobalSections_resLE, Hom.toPartialMap_hom, AlgebraicGeometry.irreducibleSpace_of_isIntegral, height_of_isClosed, map_PrimeSpectrum_basicOpen_of_affine, isoOfEq_inv, AlgebraicGeometry.IsAffineOpen.isoSpec_inv_appTop, germ_stalkClosedPointTo, zeroLocus_iInf, Modules.pushforwardId_inv_app_app, AlgebraicGeometry.IsAffineOpen.map_fromSpec_assoc, AlgebraicGeometry.IsAffineOpen.isLocalization_of_eq_basicOpen, AlgebraicGeometry.Etale.etale_appLE, Hom.opensRange_pullbackSnd, instIrreducibleSpaceCarrierCarrierCommRingCatIrreducibleComponent, AffineZariskiSite.instFaithfulOpensToOpensFunctor, Hom.germ_stalkMap_assoc, AlgebraicGeometry.LocallyQuasiFinite.quasiFinite_appLE, AlgebraicGeometry.instCompactSpaceCarrierCarrierCommRingCatEqualizerSchemeOfQuasiSeparatedSpace, Spec_map_stalkMap_fromSpecStalk, Hom.image_preimage_eq_opensRange_inter, Cover.LocallyDirected.directed, Modules.instFaithfulPresheafOfModulesToPresheafOfModules, AlgebraicGeometry.GeometricallyIrreducible.geometrically_irreducibleSpace, IsQuasiAffine.toIsImmersion, AlgebraicGeometry.QuasiCompact.isCompact_preimage, ofRestrict_appIso, AlgebraicGeometry.iSup_affineOpens_eq_top, AlgebraicGeometry.isAffineOpen_top, Pullback.Triplet.mk'_s, Hom.app_invApp'_assoc, Hom.denseRange, isEmpty_pullback_iff, AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv, Hom.quasiFiniteLocus_comp, app_eq, Hom.closePoints_subset_preimage_closedPoints, support_nilradical, AlgebraicGeometry.opensCone_pt, Cover.LocallyDirected.ofIsBasisOpensRange_le_iff, AlgebraicGeometry.LocallyOfFiniteType.finiteType_of_affine_subset, zeroLocus_empty_eq_univ, AlgebraicGeometry.mono_pushoutSection_of_iSup_eq, AlgebraicGeometry.coprodSpec_apply, AlgebraicGeometry.IsAffineOpen.isCompact, AlgebraicGeometry.morphismRestrict_base_coe, Hom.fiberHomeo_apply, AlgebraicGeometry.surjectiveOnStalks_iff, restrict_toPresheafedSpace, RingHom.IsStableUnderBaseChange.pullback_fst_appTop, AlgebraicGeometry.instCompactSpaceCarrierCarrierCommRingCatPullbackSchemeOfQuasiCompact, Hom.instIsContinuousOpensOpensFunctorGrothendieckTopologyCarrierCarrierCommRingCat, AlgebraicGeometry.instIsDomainCarrierStalkCommRingCatPresheafOfIsIntegral, AlgebraicGeometry.IsAffineOpen.map_fromSpec, germ_stalkClosedPointTo_Spec_fromSpecStalk_assoc, affineBasisCover_is_basis, AlgebraicGeometry.IsLocallyArtinian.isArtinianRing_presheaf_obj, Opens.toSpecΓ_naturality, Opens.toSpecΓ_SpecMap_presheaf_map_top, AlgebraicGeometry.range_eq_range_of_surjective, comp_toLRSHom_assoc, restrictFunctorΓ_inv_app, Cover.isOpenMap_fromGlued, Modules.restrictAdjunction_counit_app_app, AlgebraicGeometry.instGeometricallyIrreducibleMorphismRestrict, AlgebraicGeometry.instIsDomainCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTopOfIsIntegral, Pullback.Triplet.ofPoint_x, Opens.mem_ι_image_iff, AlgebraicGeometry.Proj.fromOfGlobalSections_toSpecZero_assoc, Hom.exists_isIso_morphismRestrict_toNormalization, AlgebraicGeometry.AffineSpace.toSpecMvPolyIntEquiv_symm_apply, AlgebraicGeometry.morphismRestrict_ι_assoc, AlgebraicGeometry.functionField_isFractionRing_of_isAffineOpen, IdealSheafData.ideal_sSup, Hom.stalkMap_congr_point, AlgebraicGeometry.ValuativeCriterion.Existence.specializingMap, fromSpecStalk_toSpecΓ_assoc, Modules.pushforwardCongr_hom_app_app, AlgebraicGeometry.AffineSpace.SpecIso_hom_appTop, AlgebraicGeometry.exists_isAffineOpen_preimage_eq, AlgebraicGeometry.IsAffineOpen.iSup_of_disjoint, Modules.pushforward_obj_obj, Hom.range_fiberι, Hom.preimage_bot, Cover.iUnion_range, mem_basicOpen', AlgebraicGeometry.instPrespectralSpaceCarrierCarrierCommRingCat, AlgebraicGeometry.pullbackRestrictIsoRestrict_hom_morphismRestrict_assoc, Hom.mem_opensRange, zeroLocus_eq_univ_iff_subset_nilradical, Opens.topIso_inv, SpecToEquivOfField_eq_iff, AlgebraicGeometry.isCompact_iff_exists, AlgebraicGeometry.IsAffineOpen.isCompact_pullback_inf, Hom.opensFunctor_map_homOfLE, Modules.instPreservesLimitsPresheafAbCarrierCommRingCatToPresheaf, AlgebraicGeometry.Spec_presheaf, RationalMap.dense_domain, AlgebraicGeometry.HasRingHomProperty.stalkMap, Hom.resLE_eq_morphismRestrict, Hom.iUnion_support_ker_openCover_map_comp, inv_app, Hom.id_appTop, AlgebraicGeometry.isAffineHom_diagonal_iff, AlgebraicGeometry.Proj.basicOpenIsoAway_hom, AlgebraicGeometry.image_morphismRestrict_preimage, IdealSheafData.gc, emptyTo_base, AlgebraicGeometry.Proj.map_preimage_basicOpen, zeroLocus_mul, AlgebraicGeometry.IsOpenImmersion.comp_lift_assoc, OpenCover.finiteSubcover_f, stalkSpecializes_stalkMap, Modules.germ_restrictStalkNatIso_inv_app, Spec_map_stalkSpecializes_fromSpecStalk, AlgebraicGeometry.SpecMap_preimage_basicOpen, AlgebraicGeometry.isField_stalk_of_closure_mem_irreducibleComponents, ideal_ker_le_ker_ΓSpecIso_inv_comp, stalkMap_congr_assoc, AlgebraicGeometry.instIsGermInjectiveAtCoeContinuousMapCarrierCarrierCommRingCatHomTopCatBaseOfIsOpenImmersion, Opens.toSpecΓ_SpecMap_map, isoSpec_inv_toSpecΓ, Pullback.SpecTensorTo_SpecOfPoint, Hom.image_le_image_of_le, ofRestrict_toLRSHom_base, hom_inv_apply, Hom.range_asFiberHom, isEmpty_of_commSq, AlgebraicGeometry.isIso_ΓSpec_adjunction_unit_app_basicOpen, AlgebraicGeometry.isNoetherian_iff_of_finite_affine_openCover, Hom.tendsto_cofinite_cofinite, ι_toIso_inv, instIsOverFromSpecStalkOfMem, Hom.preimage_le_preimage_of_le, Hom.stalkSpecializes_stalkMap, Hom.coe_resLE_base, AlgebraicGeometry.IsAffineOpen.fromSpecStalk_closedPoint, AlgebraicGeometry.isPullback_opens_inf, AlgebraicGeometry.SmoothOfRelativeDimension.exists_isStandardSmoothOfRelativeDimension, AlgebraicGeometry.noetherianSpace_of_isAffine, AlgebraicGeometry.exists_pow_mul_eq_zero_of_res_basicOpen_eq_zero_of_isCompact, Hom.preimageIso_hom_ι_assoc, IdealSheafData.ideal_le_comap_ideal, AlgebraicGeometry.IsClosedImmersion.base_closed, AlgebraicGeometry.quasiCompact_iff, IdealSheafData.equivOfIsAffine_apply, AlgebraicGeometry.isPullback_fiberToSpecResidueField_of_isPullback, AlgebraicGeometry.isEmpty_of_commSq_sigmaι_of_ne, Modules.Hom.zero_app, IdealSheafData.equivOfIsAffine_symm_apply, congr_app, IdealSheafData.ideal_mul, AlgebraicGeometry.IsZariskiLocalAtTarget.restrict, Opens.stalkIso_inv, SpecΓIdentity_hom_app, AlgebraicGeometry.locallyQuasiFinite_iff, preimage_comp, Hom.stalkMap_congr_hom_assoc, Cover.RelativeGluingData.toBase_preimage_eq_opensRange_ι, IdealSheafData.support_mul, AlgebraicGeometry.sourceLocalClosure.iff_forall_exists, AlgebraicGeometry.IsFinite.instMorphismRestrict, Pullback.Triplet.isPullback_SpecMap_tensor, forget_map, AlgebraicGeometry.sigmaMk_mk, AlgebraicGeometry.affineLocally_iff_forall_isAffineOpen, Pullback.Triplet.specTensorTo_base_snd, Hom.formallyUnramified_appLE, PartialMap.fromSpecStalkOfMem_compHom, isoSpec_inv_preimage_zeroLocus, AlgebraicGeometry.effectiveEpi_base_of_flat, AlgebraicGeometry.ProjectiveSpectrum.Proj.toOpen_toSpec_val_c_app_assoc, AlgebraicGeometry.IsPreimmersion.isEmbedding, AlgebraicGeometry.functionField_isScalarTower, Hom.irreducibleComponentsEquiv_symm_apply_coe, Modules.toOpen_fromTildeΓ_app, Hom.SpecMap_residueFieldMap_fromSpecResidueField_assoc, exists_germ_injective, forgetToTop_map, AlgebraicGeometry.IsOpenImmersion.iff_isIso_stalkMap, Cover.instIsIsoCommRingCatStalkMapFromGlued, Hom.preimage_top, coprodPresheafObjIso_hom_snd_assoc, Hom.toNormalization_app_preimage, Pullback.carrierEquiv_eq_iff, Hom.toImage_app_injective, isoSpec_hom, AlgebraicGeometry.instHasAffinePropertyQuasiSeparatedQuasiSeparatedSpaceCarrierCarrierCommRingCat, ι_toIso_inv_assoc, Hom.id_preimage, AlgebraicGeometry.Flat.instMorphismRestrict, AlgebraicGeometry.IsAffineOpen.opensRange_fromSpec, germ_stalkClosedPointTo_Spec_fromSpecStalk, IdealSheafData.ideal_le_ker_glueDataObjι, AlgebraicGeometry.SpecToEquivOfLocalRing_apply_snd_coe, AlgebraicGeometry.instQuasiCompactMorphismRestrict, forget_obj, eqToHom_c_app, Hom.resLE_app_top, IdealSheafData.coe_support_mkOfMemSupportIff, AlgebraicGeometry.IsAffineOpen.self_le_iSup_basicOpen_iff, zeroLocus_biInf, exists_affine_mem_range_and_range_subset, AlgebraicGeometry.Proj.basicOpenToSpec_app_top, Hom.image_preimage_le, AffineZariskiSite.generate_presieveOfSections, Hom.image_top_eq_opensRange, forgetToTop_obj, AlgebraicGeometry.IsOpenImmersion.isOpen_range, affineOpenCover_idx, Opens.isoOfLE_hom_ι_assoc, Hom.fromNormalization_preimage, Hom.iSup_preimage_eq_top, toSpecΓ_naturality_assoc, Pullback.Triplet.snd_SpecTensorTo_apply, Modules.pushforwardComp_inv_app_app, AlgebraicGeometry.SurjectiveOnStalks.iff_of_isAffine, AlgebraicGeometry.SpecMap_ΓSpecIso_hom, compactSpace_of_isAffine, AlgebraicGeometry.IsAffineOpen.isLocalization_basicOpen, zeroLocus_iUnion, AlgebraicGeometry.IsReduced.component_reduced, AlgebraicGeometry.IsIntegralHom.isIntegral_app, ker_ideal_of_isPullback_of_isOpenImmersion, presieve₀_mem_precoverage_iff, comp_base, AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv_appTop_coord, Pullback.instNonemptyCarrierCarrierCommRingCatPullbackOfSubsingleton, Hom.image_preimage_eq_opensRange_inf, AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv_over_assoc, AlgebraicGeometry.isIso_iff_isIso_stalkMap, AlgebraicGeometry.IsAffineOpen.stalkMap_injective, Hom.appLE_map', Hom.image_iSup₂, isoSpec_hom_naturality, zeroLocus_singleton, AlgebraicGeometry.instIsEmptyCarrierCarrierCommRingCatEmptyCollectionScheme, toSpecΓ_isoSpec_inv_assoc, Pullback.residueFieldCongr_inv_residueFieldMap_ofPoint, IdealSheafData.ideal_map, AlgebraicGeometry.IsOpenImmersion.range_pullbackSnd, AlgebraicGeometry.isLocallyArtinian_iff_of_isOpenCover, AlgebraicGeometry.IsImmersion.instMorphismRestrict, Hom.resLE_map, Hom.isDiscrete_preimage_singleton, Hom.comp_app, IdealSheafData.range_glueDataObjι_ι_eq_support_inter, hom_base_inv_base, AlgebraicGeometry.isNoetherian_iff, AlgebraicGeometry.continuousMapPresheaf_map, AlgebraicGeometry.IsOpenImmersion.range_pullback_fst_of_right, AlgebraicGeometry.Proj.ι_comp_map, toSpecΓ_naturality, Hom.coe_resLE_apply, Hom.quasiFiniteLocus_eq_top, AlgebraicGeometry.pointEquivClosedPoint_apply_coe, AlgebraicGeometry.IsNoetherian.toCompactSpace, AlgebraicGeometry.IsIntegral.nonempty, Hom.ι_toNormalization, Hom.smooth_appLE, IdealSheafData.ideal_pow, Opens.germ_stalkIso_inv, AlgebraicGeometry.IsNoetherian.noetherianSpace, range_fromSpecStalk, homeoOfIso_symm, comp_coeBase_assoc, AlgebraicGeometry.instCompactSpaceCarrierCarrierCommRingCatPullbackSchemeOfQuasiCompact_1, AlgebraicGeometry.IsAffineOpen.ideal_le_iff, comp_appLE, AlgebraicGeometry.quasiCompact_affineProperty_iff_quasiSeparatedSpace, AlgebraicGeometry.isIso_fromTildeΓ_of_presentation, Hom.congr_app, Hom.preimage_iSup, Hom.toPartialMap_domain, AlgebraicGeometry.coprodMk_inl, Opens.toSpecΓ_naturality_assoc, ofRestrict_appLE, Hom.QuasiFiniteAt.quasiFiniteAt, Pullback.Triplet.hy, AlgebraicGeometry.IsAffineOpen.isoSpec_inv, Hom.id_app, Γ_obj, Opens.fromSpecStalkOfMem_ι_assoc, AlgebraicGeometry.isLocallyNoetherian_iff_of_affine_openCover, IdealSheafData.instIsEmptyCarrierCarrierCommRingCatSubschemeTop, Pullback.carrierEquiv_symm_fst, IdealSheafData.mem_support_iff_of_mem, Hom.asFiberHom_fiberι_assoc, AlgebraicGeometry.instIsClosedUnderIsomorphismsSchemeIrreducibleSpaceCarrierCarrierCommRingCat, AlgebraicGeometry.Proj.mem_basicOpen, AlgebraicGeometry.IsLocallyArtinian.isArtinianRing_of_isAffine, Hom.germ_stalkMap_apply, preimage_basicOpen_top, ΓSpecIso_inv, Hom.app_appIso_inv_assoc, AlgebraicGeometry.IsAffineOpen.basicOpen_fromSpec_app, Opens.toSpecΓ_SpecMap_presheaf_map_assoc, AlgebraicGeometry.quasiCompact_over_affine_iff, opensRange_homOfLE, Pullback.range_fst_comp, Hom.coequifibered_normalizationDiagramMap, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toScheme_toLocallyRingedSpace, isoSpec_hom_naturality_assoc, isoSpec_inv_image_zeroLocus, IdealSheafData.range_glueDataObjι_ι, germToFunctionField_injective, Hom.isIso_base, Modules.germ_restrictStalkNatIso_hom_app, AlgebraicGeometry.isIso_morphismRestrict_iff_isIso_app, AlgebraicGeometry.Proj.iSup_basicOpen_eq_top', hom_base_inv_base_assoc, AlgebraicGeometry.morphismRestrict_base, Opens.ι_image_basicOpen_topIso_inv, zeroLocus_setMul, AlgebraicGeometry.SpecMap_ΓSpecIso_inv_toSpecΓ, Opens.nonempty_iff, ι_image_homOfLE_le_ι_image, toSpecΓ_preimage_zeroLocus, Hom.preimage_image_eq, Hom.quasiFiniteAt_comp_iff_of_isOpenImmersion, AlgebraicGeometry.isBasis_preimage_isAffineOpen, Cover.isOpenEmbedding_fromGlued, Hom.app_surjective, AlgebraicGeometry.coe_opensRestrict_symm_apply, AlgebraicGeometry.affineLocally_iff_affineOpens_le, fromSpecStalk_appTop, OpenCover.isOpenCover_opensRange, Modules.toPresheaf_obj, AlgebraicGeometry.pullbackRestrictIsoRestrict_inv_fst, Hom.isCompact_preimage_singleton, AlgebraicGeometry.HasRingHomProperty.iff_of_iSup_eq_top, Opens.toScheme_presheaf_obj, Modules.instFaithfulPresheafAbCarrierCommRingCatToPresheaf, IdealSheafData.vanishingIdeal_ideal, Pullback.Triplet.Spec_map_tensor_isPullback, AlgebraicGeometry.IsLocalAtSource.iff_exists_resLE, id_appTop, Cover.covers, Spec_map_presheaf_map_eqToHom, Opens.toSpecΓ_appTop_assoc, zeroLocus_mono, Hom.stalkMap_congr_point_assoc, AlgebraicGeometry.exists_eq_pow_mul_of_is_compact_of_quasi_separated_space_aux, AlgebraicGeometry.instIrreducibleSpaceCarrierCarrierCommRingCatSpecOfIsDomainCarrier, stalkMap_congr_point_assoc, GlueData.isOpen_iff, AlgebraicGeometry.IsAffineOpen.primeIdealOf_isMaximal_of_isClosed, AlgebraicGeometry.QuasiCompactCover.isCompactOpenCovered_of_isCompact, fromSpecStalk_toSpecΓ, ker_toSpecΓ, IdealSheafData.ideal_bot, ker_of_isAffine, AlgebraicGeometry.IsAffineOpen.algebraMap_Spec_obj, AlgebraicGeometry.isBasis_basicOpen, Hom.mem_preimage, Spec_map_stalkMap_fromSpecStalk_assoc, AlgebraicGeometry.isInitial_iff_isEmpty, Hom.isConstructible_preimage, AlgebraicGeometry.IsImmersion.isLocallyClosed_range, Γevaluation_naturality_apply, basicOpen_le, AlgebraicGeometry.locallyOfFiniteType_iff, AlgebraicGeometry.Flat.flat_and_surjective_iff_faithfullyFlat_of_isAffine, Opens.toSpecΓ_SpecMap_presheaf_map_top_assoc, AlgebraicGeometry.ExistsHomHomCompEqCompAux.exists_index, AlgebraicGeometry.liftCoborder_app, Modules.isSheaf, Opens.toSpecΓ_preimage_basicOpen, AlgebraicGeometry.isIntegral_appTop_of_universallyClosed, Cover.exists_lift_trans_eq, zeroLocus_def, Cover.exists_of_f_eq_f, stalkClosedPointTo_fromSpecStalk, AlgebraicGeometry.IsOpenImmersion.map_ΓIso_inv_assoc, AlgebraicGeometry.ΓSpec.adjunction_counit_app, Hom.preimage_smoothLocus_eq, AlgebraicGeometry.tilde.isoTop_hom, coe_homeoOfIso_symm, restrict_presheaf_obj, AlgebraicGeometry.IsArtinianScheme.finite, AlgebraicGeometry.instIrreducibleSpaceCarrierCarrierCommRingCatPullbackSchemeOfGeometricallyIrreducibleOfUniversallyOpen, stalkMap_hom_inv_assoc, ModuleCat.Tilde.toOpen_res, AlgebraicGeometry.IsLocallyArtinian.iff_isLocallyNoetherian_and_discreteTopology, IdealSheafData.mem_supportSet_iff, id.base, IdealSheafData.range_subschemeι, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.scheme_toScheme, Opens.toSpecΓ_appTop, IdealSheafData.mem_support_iff, IdealSheafData.le_support_iff_le_vanishingIdeal, Hom.isClosedMap, Hom.isLocallyClosed_range, AlgebraicGeometry.LocallyOfFiniteType.finiteType_appLE, AlgebraicGeometry.ProjectiveSpectrum.Proj.toStalk_stalkMap_toSpec_assoc, AlgebraicGeometry.opensDiagram_map, Hom.finitePresentation_appLE, residue_descResidueField, AlgebraicGeometry.Γ_restrict_isLocalization, AlgebraicGeometry.morphismRestrict_appLE, Hom.appIso_inv_app_presheafMap, Opens.toScheme_carrier, AlgebraicGeometry.IsClosedImmersion.iff_isPreimmersion, AlgebraicGeometry.iSup_basicOpen_of_span_eq_top, Hom.app_eq, AlgebraicGeometry.morphismRestrict_ι, Hom.comp_appLE, Modules.restrictFunctorComp_hom_app_app, Hom.preimageIso_inv_ι, SpecMap_stalkSpecializes_fromSpecStalk, descResidueField_fromSpecResidueField, AlgebraicGeometry.ProjectiveSpectrum.Proj.toStalk_stalkMap_toSpec, Hom.apply_mem_image_iff, Hom.asFiberHom_apply, Modules.restrictFunctorComp_inv_app_app, AlgebraicGeometry.ΓSpecIso_obj_hom, AlgebraicGeometry.Proj.basicOpen_mul, AlgebraicGeometry.genericPoint_eq_of_isOpenImmersion, homeoOfIso_apply, AlgebraicGeometry.Spec.germ_stalkMapIso_hom_assoc, IdealSheafData.le_ofIdeals_iff, Hom.preimage_basicOpen_top, AlgebraicGeometry.FormallyUnramified.formallyUnramified_of_affine_subset, Opens.germ_stalkIso_inv_assoc, AlgebraicGeometry.formallySmooth_stalkMap_iff, Modules.restrict_map, PartialMap.restrict_id, AlgebraicGeometry.isIso_stalkMap_coprodSpec, AlgebraicGeometry.injective_germ_basicOpen, AlgebraicGeometry.AffineSpace.spec_le_iff, IsLocallyDirected.homOfLE_tAux, AlgebraicGeometry.WeaklyEtale.instMorphismRestrict, AlgebraicGeometry.HasRingHomProperty.iff_appLE, Opens.toSpecΓ_SpecMap_map_assoc, Hom.image_mono, AlgebraicGeometry.instQuasiSoberCarrierCarrierCommRingCat, Opens.exists_toScheme, AlgebraicGeometry.IsAffineOpen.self_le_basicOpen_union_iff, AlgebraicGeometry.exists_appTop_π_eq_of_isAffine_of_isLimit, instIsIsoCommRingCatResidueFieldMapOfIsOpenImmersion, instIsOverMapResidueFieldMapOverInferInstanceOverClass, OpenCover.iSup_opensRange, AlgebraicGeometry.HasAffineProperty.restrict, AlgebraicGeometry.instT0SpaceCarrierCarrierCommRingCat, Γevaluation_naturality_assoc, AlgebraicGeometry.opensDiagramι_app, Pullback.carrierEquiv_symm_snd, IdealSheafData.coe_support_ofIdealTop, AlgebraicGeometry.isIso_pushoutSection_of_isCompact_of_flat_right_of_ringHomFlat, IdealSheafData.support_antitone, Hom.isIso_toLRSHom, component_nontrivial, AlgebraicGeometry.isCompl_range_inl_inr, zeroLocus_iInf_of_nonempty, AffineZariskiSite.generate_presieveOfSections_mem_grothendieckTopology, basicOpen_pow, Opens.toSpecΓ_top, AlgebraicGeometry.HasRingHomProperty.appLE, AlgebraicGeometry.IsLocallyNoetherian.quasiSeparatedSpace, Hom.ker_apply, IdealSheafData.ideal_comap_of_isOpenImmersion, Pullback.range_map, AlgebraicGeometry.SurjectiveOnStalks.isEmbedding_pullback, AlgebraicGeometry.AffineSpace.SpecIso_inv_appTop_coord, AlgebraicGeometry.germ_stalkClosedPointIso_hom_assoc, Hom.isSpectralMap, IdealSheafData.support_eq_top_iff, AlgebraicGeometry.mem_range_iff_of_surjective, AffineZariskiSite.restrictIsoSpec_hom_app, AlgebraicGeometry.instIsIsoSchemeMorphismRestrict, Cover.LocallyDirected.ofIsBasisOpensRange_trans, Hom.isLocallyConstructible_image, evaluation_eq_zero_iff_notMem_basicOpen, Modules.toPresheaf_map, stalkMap_germ_assoc, Hom.appIso_inv_app, Hom.liftCoborder_preimage, stalkMap_congr_hom_assoc, Opens.germ_stalkIso_hom, Hom.opensRange_pullbackFst, IdealSheafData.mem_supportSet_iff_mem_support, stalkSpecializes_stalkMap_apply, zeroLocus_radical, Modules.restrictFunctorCongr_inv_app_app, AlgebraicGeometry.pullbackRestrictIsoRestrict_hom_ι_assoc, AlgebraicGeometry.exists_map_preimage_eq_map_preimage, AlgebraicGeometry.affineOpensRestrict_apply_coe_coe, AlgebraicGeometry.compactSpace_iff_quasiCompact, Hom.map_appLE_assoc, irreducibleComponentOpen_eq_top, AffineZariskiSite.presieveOfSections_surjective, Opens.ι_appIso, AlgebraicGeometry.IsAffineOpen.isoSpec_inv_toSpecΓ, AlgebraicGeometry.disjoint_opensRange_sigmaι, comp_base_assoc, AlgebraicGeometry.instLocallyQuasiFiniteMorphismRestrict, stalkMap_id, Hom.map_appLE'_assoc, Spec.residue_residueFieldIso_hom_assoc, AlgebraicGeometry.IsAffineOpen.toSpecΓ_isoSpec_inv, Hom.appIso_inv_naturality, IdealSheafData.map_ideal, Opens.toSpecΓ_SpecMap_presheaf_map, basicOpen_add_le, Hom.comp_app_assoc, comp_app, AlgebraicGeometry.exists_of_res_zero_of_qcqs_of_top, zeroLocus_map, Hom.finite_app, AlgebraicGeometry.isIso_iff_isOpenImmersion, AffineZariskiSite.coequifibered_iff_forall_isLocalizationAway, AlgebraicGeometry.instGeometricallyReducedMorphismRestrict, AlgebraicGeometry.IsAffineOpen.fromSpec_preimage_zeroLocus, AlgebraicGeometry.exists_preimage_eq, evaluation_naturality, GlueData.ι_eq_iff, Modules.toOpen_fromTildeΓ_app_assoc, AlgebraicGeometry.FormallyUnramified.instIsSeparableCarrierResidueFieldCoeContinuousMapCarrierCarrierCommRingCatHomTopCatBaseOfLocallyOfFiniteType, Hom.ideal_ker_le, AlgebraicGeometry.AffineSpace.comp_homOfVector, AlgebraicGeometry.smooth_iff, AlgebraicGeometry.diagonal_isAffine_iff_forall_isAffineOpen_inf, Hom.appIso_hom', zeroLocus_inf, AlgebraicGeometry.AffineSpace.isOpenMap_over, AlgebraicGeometry.surjective_iff, IdealSheafData.ofIdeals_mono, instIsOpenImmersionToSpecΓOfIsQuasiAffine, homOfLE_appTop, AlgebraicGeometry.germ_injective_of_isIntegral, isoSpec_inv_naturality_assoc, AlgebraicGeometry.IsAffineOpen.fromSpec_app_self, AlgebraicGeometry.IsAffineOpen.biInf, homOfLE_homOfLE_assoc, AlgebraicGeometry.IsOpenImmersion.opensEquiv_apply_coe, AlgebraicGeometry.isClosedImmersion_iff_isAffineHom, AlgebraicGeometry.isLocalIso_iff, Hom.resLE_congr, AlgebraicGeometry.AffineSpace.hom_ext_iff, SpecMap_stalkMap_fromSpecStalk_assoc, AlgebraicGeometry.exists_isAffineOpen_mem_and_subset, Hom.preimage_basicOpen, singleton_mem_precoverage_iff, IdealSheafData.subschemeι_app, Hom.appLE_appIso_inv, AlgebraicGeometry.IsLocallyArtinian.discreteTopology, Hom.isOpenMap, PartialMap.equiv_iff_of_isSeparated, isoSpec_Spec_hom, basicOpen_res_eq, Pullback.Triplet.hx, Hom.isAffineOpen_iff_of_isOpenImmersion, Hom.app_invApp', Hom.stalkFunctor_toImage_injective, AlgebraicGeometry.isOpenImmersion_iff_stalk, AlgebraicGeometry.isNoetherian_iff_of_finite_iSup_eq_top, image_basicOpen, AlgebraicGeometry.exists_appTop_map_eq_zero_of_isAffine_of_isLimit, AlgebraicGeometry.tilde.isIso_toOpen_top, instQuasiSeparatedSpaceCarrierCarrierCommRingCatOfIsSeparated, AlgebraicGeometry.SurjectiveOnStalks.surj_on_stalks, Hom.stalkSpecializes_stalkMap_assoc, AlgebraicGeometry.HasRingHomProperty.appTop, AlgebraicGeometry.isIntegral_iff_irreducibleSpace_and_isReduced, AlgebraicGeometry.IsAffineOpen.image_of_isOpenImmersion, AlgebraicGeometry.IsClosedImmersion.Spec_map_residue, isNilpotent_iff_basicOpen_eq_bot, AlgebraicGeometry.SurjectiveOnStalks.stalkMap_surjective, AlgebraicGeometry.IsIntegral.component_integral, AlgebraicGeometry.morphismRestrict_comp, SpecMap_stalkSpecializes_fromSpecStalk_assoc, Hom.comp_appLE_assoc, AlgebraicGeometry.IsProper.instMorphismRestrict, isPullback_toSpecΓ_toSpecΓ, Hom.genericPoint_mem_smoothLocus_of_perfectField, AlgebraicGeometry.IsZariskiLocalAtTarget.iff_of_iSup_eq_top, instSubsingletonCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensBot, AlgebraicGeometry.IsArtinianScheme.toCompactSpace, ΓSpecIso_naturality, Hom.stalkMap_hom_inv, IdealSheafData.support_comap, restrictFunctorΓ_hom_app, stalkMap_congr_hom, Modules.restrictFunctorCongr_hom_app_app, IdealSheafData.vanishingIdeal_sup, Pullback.range_snd_comp, Modules.restrictFunctorId_inv_app_app, Opens.isoOfLE_hom_ι, Opens.fromSpecStalkOfMem_ι, Modules.pushforwardComp_hom_app_app, AlgebraicGeometry.instIsFractionRingCarrierStalkCommRingCatPresheafFunctionField, iso_inv_base_hom_base, AlgebraicGeometry.germ_stalkClosedPointIso_hom, AlgebraicGeometry.range_eq_univ, toSpecΓ_isoSpec_inv, AlgebraicGeometry.IsFinite.instHasAffinePropertyAndIsAffineFiniteCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTopHomAppTop, AlgebraicGeometry.SpecToEquivOfLocalRing_apply_fst, Hom.comp_apply, homOfLE_homOfLE, AlgebraicGeometry.IsAffineOpen.instIsAffineToSchemeBasicOpen, Modules.mapPresheaf_app, Hom.isOpenEmbedding, AlgebraicGeometry.opensDiagram_obj, Hom.preservesLocalization_normalizationDiagramMap, AlgebraicGeometry.Proj.basicOpen_mono, AlgebraicGeometry.tilde.instIsLocalizedModuleCarrierCarrierOfCarrierStalkAbPresheafPrimeComplAsIdealHomToStalk, AlgebraicGeometry.Proj.awayToSection_comp_appLE, residueFieldCongr_trans, AlgebraicGeometry.Spec.map_apply, AlgebraicGeometry.pointOfClosedPoint_apply, AlgebraicGeometry.IsAffineOpen.basicOpen_basicOpen_is_basicOpen, toIso_inv_ι_assoc, AlgebraicGeometry.morphismRestrict_app', evaluation_naturality_apply, Hom.isEmbedding, AlgebraicGeometry.sigmaι_eq_iff, AlgebraicGeometry.opensCone_π_app, toSpecΓ_appTop, Hom.appIso_inv_naturality_assoc, Hom.appLE_map_assoc, AlgebraicGeometry.IsOpenImmersion.opensEquiv_symm_apply, IdealSheafData.support_eq_bot_iff, irreducibleComponentι_apply, AlgebraicGeometry.GeometricallyIrreducible.irreducibleSpace_of_subsingleton, stalkMap_comp, residue_surjective, AlgebraicGeometry.isCompactOpen_iff_eq_finset_affine_union, AlgebraicGeometry.IsAffineOpen.iSup_basicOpen_eq_self_iff, Hom.comp_appIso, AlgebraicGeometry.instJacobsonSpaceCarrierCarrierCommRingCatSpecOfOfIsJacobsonRing, basicOpen_one, AlgebraicGeometry.IsOpenImmersion.range_pullback_to_base_of_left, AlgebraicGeometry.over_def, Hom.appIso_inv_appLE_assoc, AlgebraicGeometry.Proj.homOfLE_toBasicOpenOfGlobalSections_ι_assoc, Modules.Hom.id_app, IdealSheafData.map_vanishingIdeal, Hom.stalkMap_inv_hom_assoc, AlgebraicGeometry.targetAffineLocally_affineAnd_iff, Hom.appIso_inv_app_assoc, AlgebraicGeometry.IsAffineOpen.fromSpec_preimage_self, Opens.toSpecΓ_SpecMap_appLE_assoc, Hom.ker_eq_top_iff_isEmpty, AlgebraicGeometry.isClosed_singleton_iff_locallyOfFiniteType, Cover.ulift_I₀, zeroLocus_eq_univ_iff_subset_nilradical_of_isCompact, Modules.Hom.isIso_iff_isIso_app, Hom.residueFieldMap_congr, AlgebraicGeometry.Spec_zeroLocus_eq_zeroLocus, Opens.ι_appTop, Modules.Hom.comp_app, Hom.map_mem_image_iff, IdealSheafData.support_iSup, coprodPresheafObjIso_hom_snd, map_basicOpen, AlgebraicGeometry.IsOpenImmersion.affineOpensEquiv_apply_coe_coe, AlgebraicGeometry.coprodSpec_coprodMk, Cover.exists_of_trans_eq_trans, IdealSheafData.opensRange_subschemeCover_map, AlgebraicGeometry.isRetrocompact_basicOpen, Hom.preimageIso_hom_ι, AlgebraicGeometry.exists_affineOpens_le_appLE_of_appLE, AlgebraicGeometry.affinePreimage, Hom.injective, AlgebraicGeometry.isIso_iff_isOpenImmersion_and_epi_base, AlgebraicGeometry.mono_pushoutSection_of_isCompact_of_flat_left_of_ringHomFlat, instIsOverMapStalkSpecializesCommRingCatPresheaf, AlgebraicGeometry.quasiSeparatedSpace_of_isAffine, isoSpec_inv_naturality, AlgebraicGeometry.IsAffineOpen.toSpecΓ_fromSpec_assoc, AlgebraicGeometry.Spec_carrier, AlgebraicGeometry.isArtinianScheme_iff, Hom.id_appIso, empty_carrier_carrier, AlgebraicGeometry.IsArtinianScheme.iff_isNoetherian_and_discreteTopology, Hom.preimageIso_inv_ι_assoc, Hom.quasiFiniteLocus_eq_top_iff, AlgebraicGeometry.instQuasiSeparatedToSpecΓOfQuasiSeparatedSpaceCarrierCarrierCommRingCat, AlgebraicGeometry.IsAffineOpen.basicOpenSectionsToAffine_isIso, AlgebraicGeometry.isAffineHom_iff, Hom.asFiberHom_fiberι, AlgebraicGeometry.exists_smooth_of_formallySmooth_stalk, comp_appTop_assoc, AlgebraicGeometry.QuasiCompact.compactSpace_of_compactSpace, local_affine, AlgebraicGeometry.Proj.basicOpen_one, Hom.resLE_comp_resLE_assoc, AlgebraicGeometry.isIso_pushoutSection_of_iSup_eq, IdealSheafData.ideal_top, AlgebraicGeometry.quasiSeparated_iff_quasiSeparatedSpace, AlgebraicGeometry.LocallyOfFiniteType.stalkMap, Hom.etale_appLE, AlgebraicGeometry.locallyQuasiFinite_iff_isDiscrete_preimage_singleton, IsLocallyDirected.ι_eq_ι_iff, isoSpec_Spec, OpenCover.compactSpace, AlgebraicGeometry.instLocallyOfFiniteTypeMorphismRestrict, Hom.appLE_congr, AlgebraicGeometry.IsIntegralHom.instMorphismRestrict, IdealSheafData.isLocalization_away, AlgebraicGeometry.isCompact_and_isOpen_iff_finite_and_eq_biUnion_affineOpens, Pullback.Triplet.specTensorTo_base_fst, ofRestrict_toLRSHom_c_app, AlgebraicGeometry.compactSpace_iff_exists, AlgebraicGeometry.instIrreducibleSpaceCarrierCarrierCommRingCatFiberOfGeometricallyIrreducible, Hom.preimage_sup, AlgebraicGeometry.IsClosedImmersion.hasAffineProperty, preimage_opensRange_toSpecΓ, AlgebraicGeometry.instIsOpenImmersionMorphismRestrict, AlgebraicGeometry.Proj.fromOfGlobalSections_toSpecZero, Spec.algebraMap_residueFieldIso_inv, iso_hom_base_inv_base, stalkMap_congr_point, AlgebraicGeometry.IsAffineOpen.exists_basicOpen_le, eq_zeroLocus_of_isClosed_of_isAffine, Opens.ι_preimage_self, AlgebraicGeometry.IsAffineOpen.comap_primeIdealOf_appLE, stalkMap_hom_inv, residueFieldCongr_refl, Hom.appLE_appIso_inv_assoc, toSpecΓ_image_zeroLocus, AlgebraicGeometry.instQuasiSeparatedMorphismRestrict, AlgebraicGeometry.HasAffineProperty.iff_of_iSup_eq_top, Hom.image_iSup, AlgebraicGeometry.isImmersion_iff, AlgebraicGeometry.quasiSeparatedSpace_iff_forall_affineOpens, AlgebraicGeometry.instNonemptyCarrierCarrierCommRingCatSpecOfNontrivialCarrier, AlgebraicGeometry.toSpecΓ_SpecMap_ΓSpecIso_inv_assoc, homOfLE_app, Hom.comp_preimage, Modules.instIsIsoAbApp, Hom.naturality_assoc, IdealSheafData.opensRange_glueDataObjMap, AlgebraicGeometry.IsAffineOpen.fromSpec_app_self_apply, stalkMap_hom_inv_apply, Hom.iInf_ker_openCover_map_comp_apply, Hom.isOpen_quasiFiniteAt, AlgebraicGeometry.eq_zero_of_basicOpen_eq_bot, homOfLE_appLE, inv_appTop, AlgebraicGeometry.isLocalization_away_of_isAffine, Hom.isIso_app, AlgebraicGeometry.IsAffineOpen.ι_basicOpen_preimage, basicOpen_appLE, Hom.quasiFiniteAt_iff, Hom.stalkSpecializes_stalkMap_apply, Spec_map_stalkSpecializes_fromSpecStalk_assoc, AlgebraicGeometry.morphismRestrict_appTop, Hom.exists_mem_and_isIso_morphismRestrict_toNormalization, IdealSheafData.ofIdealTop_ideal, AlgebraicGeometry.GeometricallyIrreducible.eq_geometrically, AlgebraicGeometry.Proj.basicOpen_eq_iSup_proj, AlgebraicGeometry.isOpenMap_of_generalizingMap, Hom.preimage_opensRange, AlgebraicGeometry.ΓSpecIso_hom_stalkClosedPointIso_inv, AlgebraicGeometry.AffineSpace.toSpecMvPolyIntEquiv_apply, AlgebraicGeometry.Proj.isBasis_basicOpen, descResidueField_stalkClosedPointTo_fromSpecResidueField, AlgebraicGeometry.IsAffineOpen.fromSpec_preimage_basicOpen, AlgebraicGeometry.Flat.flat_of_affine_subset, AlgebraicGeometry.isCompactOpen_iff_eq_basicOpen_union, Hom.comp_toLRSHom, AlgebraicGeometry.Spec.algebraMap_stalkIso_inv, AlgebraicGeometry.geometricallyIrreducible_iff, AlgebraicGeometry.isDominant_iff, AlgebraicGeometry.Flat.generalizingMap, Hom.fiberι_asFiber, AlgebraicGeometry.AffineSpace.isIntegralHom_over_iff_isEmpty, AlgebraicGeometry.HasAffineProperty.affineAnd_iff, default_asIdeal, AlgebraicGeometry.IsOpenImmersion.app_eq_appIso_inv_app_of_comp_eq, Hom.id_base, IdealSheafData.ker_subschemeι_app, AlgebraicGeometry.IsOpenImmersion.ΓIso_inv, Modules.Hom.sub_app, AlgebraicGeometry.eq_top_of_sigmaSpec_subset_of_isCompact, AlgebraicGeometry.IsOpenImmersion.range_pullback_snd_of_left, AlgebraicGeometry.IsOpenImmersion.instIsIsoCommRingCatStalkMap, AlgebraicGeometry.isPreimmersion_iff, AffineCover.covers, AlgebraicGeometry.ΓSpec.adjunction_homEquiv_apply, Opens.isoOfLE_inv_ι, Hom.ι_toNormalization_assoc, Hom.Spec_map_residueFieldMap_fromSpecResidueField_assoc, isLocalHom_stalkClosedPointTo', AlgebraicGeometry.IsFinite.finite_preimage_singleton, AlgebraicGeometry.IsLocalAtTarget.restrict, Hom.fromNormalization_app_assoc, isoSpec_inv_toSpecΓ_assoc, AlgebraicGeometry.Proj.fromOfGlobalSections_morphismRestrict, IdealSheafData.ideal_iInf, AlgebraicGeometry.IsAffineOpen.primeIdealOf_eq_map_closedPoint, Hom.isoImage_inv_ι, AlgebraicGeometry.stalkMap_injective_of_isOpenMap_of_injective, GlueData.ι_isoCarrier_inv, stalkMap_germ_apply, IdealSheafData.range_glueDataObjι, Hom.isoImage_inv_ι_assoc, Hom.instIsIsoCommRingCatAppObjOpensOpensFunctor, ker_morphismRestrict_ideal, Modules.pushforwardId_hom_app_app, AlgebraicGeometry.pullbackRestrictIsoRestrict_hom_ι, Hom.germ_stalkMap, coprodPresheafObjIso_hom_fst, toOpen_eq, AlgebraicGeometry.IsOpenImmersion.range_pullback_to_base_of_right, Modules.restrict_obj, AlgebraicGeometry.IsOpenImmersion.app_ΓIso_hom_assoc, Hom.preimage_iSup_eq_top, Modules.restrictAdjunction_unit_app_app, AlgebraicGeometry.IsLocalAtTarget.iff_of_iSup_eq_top, AlgebraicGeometry.isCompl_opensRange_inl_inr, Hom.isIntegral_app, Hom.app_eq_appLE, Pullback.Triplet.fst_SpecTensorTo_apply, AlgebraicGeometry.ΓSpecIso_inv_ΓSpec_adjunction_homEquiv, Modules.Hom.add_app, IdealSheafData.isClosed_supportSet, AlgebraicGeometry.toSpecΓ_SpecMap_ΓSpecIso_inv, AlgebraicGeometry.IsOpenImmersion.map_ΓIso_inv, AlgebraicGeometry.exists_map_preimage_le_map_preimage, AlgebraicGeometry.isBasis_affine_open, AlgebraicGeometry.instIsIsoModulesSpecOfCarrierFromTildeΓUnitOpensCarrierCarrierCommRingCatRingCatSheaf, AlgebraicGeometry.IsClosedImmersion.isClosedEmbedding, Hom.map_appLE', Hom.inv_appTop, AffineZariskiSite.opensRange_relativeGluingData_map, AlgebraicGeometry.Flat.flat_appLE, AlgebraicGeometry.IsLocallyArtinian.discreteTopology_of_isAffine, AlgebraicGeometry.isIntegralHom_iff, AlgebraicGeometry.stalkMap_injective_of_isAffine, AlgebraicGeometry.Spec_zeroLocus, AlgebraicGeometry.LocallyOfFinitePresentation.finitePresentation_appLE, basicOpen_mul, ΓSpecIso_inv_naturality, IdealSheafData.coe_support_eq_eq_iInter_zeroLocus, IsJointlySurjectivePreserving.exists_preimage_snd_triplet_of_prop, AlgebraicGeometry.exists_of_res_eq_of_qcqs_of_top, Hom.finiteType_appLE, AlgebraicGeometry.exists_app_map_eq_map_of_isLimit, AffineZariskiSite.instIsDenseSubsiteOpensCarrierCarrierCommRingCatGrothendieckTopologyGrothendieckTopologyToOpensFunctor, PartialMap.mem_domain_ofFromSpecStalk, zeroLocus_biInf_of_nonempty, AlgebraicGeometry.Flat.stalkMap, Pullback.range_snd, Opens.germ_stalkIso_hom_assoc, Hom.stalkMap_comp, SpecΓIdentity_inv_app, Pullback.diagonalCoverDiagonalRange_eq_top_of_injective, AlgebraicGeometry.instJacobsonSpaceCarrierCarrierCommRingCatFiberOfLocallyOfFiniteType, Cover.exists_eq, AlgebraicGeometry.IsAffineOpen.biSup_of_disjoint, AlgebraicGeometry.IsOpenImmersion.app_ΓIso_hom, ofArrows_mem_precoverage_iff, AlgebraicGeometry.mono_pushoutSection_of_isCompact_of_flat_right, AlgebraicGeometry.isLocallyNoetherian_iff_of_iSup_eq_top, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toSchemeHom_toPshHom, IsLocallyDirected.ι_jointly_surjective, AlgebraicGeometry.basicOpen_eq_of_affine', AlgebraicGeometry.IsPreimmersion.base_embedding, Hom.coe_image, AlgebraicGeometry.AffineSpace.isoOfIsAffine_hom_appTop, Hom.coe_preimage, Opens.fromSpecStalkOfMem_toSpecΓ, Opens.toScheme_presheaf_map, AlgebraicGeometry.instIsDominantToSpecΓOfCompactSpaceCarrierCarrierCommRingCat, preimage_eq_top_of_closedPoint_mem, isoSpec_Spec_inv, IdealSheafData.supportSet_subset_zeroLocus, Hom.isClosedEmbedding, Hom.comp_toLRSHom_assoc, AlgebraicGeometry.Proj.awayι_preimage_basicOpen, Hom.quasiFiniteAt_iff_isOpen_singleton_asFiber, AlgebraicGeometry.isClosed_singleton_iff_isClosedImmersion, AlgebraicGeometry.IsFinite.finite_app, AlgebraicGeometry.instSubsingletonCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensOfIsEmpty, AlgebraicGeometry.exists_eq_pow_mul_of_isAffineOpen, Opens.forall_toScheme, germ_stalkClosedPointTo_assoc, topIso_hom, AlgebraicGeometry.Proj.awayMap_awayToSection, Hom.finite_preimage_singleton, AlgebraicGeometry.IsAffineOpen.inf, Hom.map_resLE, AlgebraicGeometry.finite_appTop_of_universallyClosed, AffineZariskiSite.toOpens_mono, AlgebraicGeometry.IsLocallyNoetherian.component_noetherian, inv_hom_apply, AlgebraicGeometry.instIsPreimmersionFromSpecStalk, AlgebraicGeometry.exists_mem_of_isClosed_of_nonempty', AlgebraicGeometry.IsIntegralHom.integral_app, Hom.resLE_preimage, residue_residueFieldMap_assoc, AlgebraicGeometry.mono_pushoutSection_of_isCompact_of_flat_right_of_ringHomFlat, IdealSheafData.support_map, AlgebraicGeometry.FormallyUnramified.stalkMap, AlgebraicGeometry.specTargetImageFactorization_app_injective, comp_toLRSHom, IsGermInjectiveAt.cond, Hom.le_preimage_resLE_iff, IdealSheafData.zeroLocus_inter_subset_supportSet, AlgebraicGeometry.IsAffineOpen.preimage_of_isIso, AlgebraicGeometry.instIrreducibleSpaceCarrierCarrierCommRingCatPullbackSchemeOfGeometricallyIrreducibleOfUniversallyOpen_1, AlgebraicGeometry.instIsClosedImmersionMorphismRestrict, Hom.preimage_inf, AlgebraicGeometry.isPullback_morphismRestrict, IdealSheafData.map_ideal', residue_residueFieldCongr, Hom.appIso_hom, OpenCover.finiteSubcover_X, AlgebraicGeometry.quasiSeparatedSpace_of_quasiSeparated, Pullback.range_fst, AlgebraicGeometry.IsOpenImmersion.iff_stalk_iso, Cover.fromGlued_isOpenEmbedding, AlgebraicGeometry.IsOpenImmersion.app_ΓIso_hom_apply, PartialMap.restrict_restrict_hom, residue_residueFieldMap, IdealSheafData.ideal_sup, evaluation_naturality_assoc, AlgebraicGeometry.AffineSpace.isoOfIsAffine_hom, AlgebraicGeometry.tilde.toOpen_res, AlgebraicGeometry.tilde.toOpen_res_assoc, Hom.ι_fromNormalization_assoc, emptyTo_c_app, stalkMap_inv_hom_apply, AlgebraicGeometry.instCompactSpaceCarrierCarrierCommRingCatFiberOfQuasiCompact, IsQuasiAffine.toCompactSpace, Hom.map_resLE_assoc, AlgebraicGeometry.IsLocalIso.exists_isOpenImmersion, Opens.mem_basicOpen_toScheme, Hom.comp_base_assoc, AlgebraicGeometry.affineAnd_apply, AlgebraicGeometry.IsAffineOpen.preimage_of_isOpenImmersion, SpecMap_residue_apply, AlgebraicGeometry.tilde.isUnit_algebraMap_end_basicOpen, AlgebraicGeometry.HasRingHomProperty.iff_exists_appLE_locally, Hom.image_le_opensRange, AlgebraicGeometry.noetherianSpace_of_isAffineOpen, Pullback.range_diagonal_subset_diagonalCoverDiagonalRange, AlgebraicGeometry.targetAffineLocally_affineAnd_iff', AlgebraicGeometry.AffineSpace.reindex_appTop_coord, AlgebraicGeometry.Spec.algebraMap_stalkIso_inv_assoc, AlgebraicGeometry.smoothOfRelativeDimension_iff, AlgebraicGeometry.quasiCompact_iff_forall_isAffineOpen, AlgebraicGeometry.HasRingHomProperty.iff_exists_appLE, instIsPreimmersionMapResidue, Hom.mem_smoothLocus, AlgebraicGeometry.ΓSpec.adjunction_homEquiv_symm_apply, AlgebraicGeometry.instSmoothMorphismRestrict, Hom.appLE_appIso_inv_apply, IdealSheafData.ideal_mono, Hom.appLE_map'_assoc, empty_presheaf, AlgebraicGeometry.IsAffineOpen.isoSpec_hom_apply, Opens.ι_app_self, PartialMap.restrict_restrict, zeroLocus_span, IdealSheafData.ideal_iSup, Hom.naturality, AlgebraicGeometry.quasiSeparatedSpace_iff_quasiSeparated, Hom.image_injective, AlgebraicGeometry.exists_appTop_π_eq_of_isLimit, AlgebraicGeometry.exists_isFinite_morphismRestrict_of_finite_preimage_singleton, AlgebraicGeometry.IsAffine.affine, AlgebraicGeometry.IsAffineOpen.instAwayCarrierObjOppositeOpensCarrierCarrierCommRingCatSpecPresheafOpOpensBasicOpen, AlgebraicGeometry.Spec_toLocallyRingedSpace, AlgebraicGeometry.AffineTargetMorphismProperty.IsLocal.to_basicOpen, Pullback.Triplet.Spec_ofPointTensor_SpecTensorTo, IdealSheafData.vanishingIdeal_top, restrict_presheaf_map, AlgebraicGeometry.Proj.basicOpen_zero, AlgebraicGeometry.formallyUnramified_iff, AlgebraicGeometry.IsAffineOpen.fromSpecStalk_isPreimmersion, Pullback.Triplet.ofPoint_s, mem_basicOpen_top, AlgebraicGeometry.QuasiCompactCover.isCompactOpenCovered_of_isAffineOpen, Opens.range_ι, AlgebraicGeometry.IsAffineOpen.range_fromSpec, AlgebraicGeometry.IsOpenImmersion.image_preimage_eq_preimage_image_of_isPullback, AlgebraicGeometry.isClosedImmersion_iff, Opens.ι_image_basicOpen', AlgebraicGeometry.Spec_closedPoint, Hom.range_subset_ker_support, AlgebraicGeometry.IsAffineOpen.isoSpec_inv_toSpecΓ_assoc, isNilpotent_of_isNilpotent_cover, AlgebraicGeometry.AffineSpace.homOverEquiv_apply, Hom.finite_preimage, Opens.ι_apply, Modules.instFullPresheafOfModulesToPresheafOfModules, Hom.dense_smoothLocus_of_perfectField, AlgebraicGeometry.quasiCompact_iff_compactSpace, AlgebraicGeometry.pullbackRestrictIsoRestrict_inv_fst_assoc, GlueData.ι_isoLocallyRingedSpace_inv, basicOpen_res, iso_inv_base_hom_base_apply, Opens.fromSpecStalkOfMem_toSpecΓ_assoc, Hom.continuous, Hom.asFiberHom_fiberToSpecResidueField, AlgebraicGeometry.Smooth.exists_isStandardSmooth, Pullback.Triplet.exists_preimage, comp_appTop, Hom.isConstructible_image, Hom.stalkMap_hom_inv_apply, AlgebraicGeometry.Surjective.surj, IdealSheafData.radical_ideal, AlgebraicGeometry.isIso_pushoutSection_of_isQuasiSeparated_of_flat_left, AlgebraicGeometry.isCompact_iff_finite_and_eq_biUnion_affineOpens, AlgebraicGeometry.IsClosedImmersion.isAffine_surjective_of_isAffine, AffineZariskiSite.mem_grothendieckTopology, stalkMap_inv_hom, AlgebraicGeometry.germ_eq_zero_of_pow_mul_eq_zero, AlgebraicGeometry.isIso_pushoutSection_iff, comp_base_apply, fromSpecStalk_app, AlgebraicGeometry.isGermInjectiveAt_iff_of_isOpenImmersion, AlgebraicGeometry.IsAffineOpen.isLocalization_stalk, AlgebraicGeometry.Smooth.smooth_appLE, Hom.irreducibleComponentsEquiv_apply_coe, AlgebraicGeometry.AffineSpace.toSpecMvPolyIntEquiv_comp, AlgebraicGeometry.exists_basicOpen_le_appLE_of_appLE_of_isAffine, AlgebraicGeometry.isField_of_universallyClosed, AlgebraicGeometry.IsAffineOpen.isoSpec_hom_base_apply, IdealSheafData.ideal_inf, AlgebraicGeometry.exists_of_res_zero_of_qcqs, Spec.algebraMap_residueFieldIso_inv_assoc, nonempty_of_isLimit, AlgebraicGeometry.exists_pow_mul_eq_zero_of_res_basicOpen_eq_zero_of_isAffineOpen, AlgebraicGeometry.IsDominant.denseRange, homOfLE_base, AlgebraicGeometry.instGeometricallyIntegralMorphismRestrict, AlgebraicGeometry.Flat.iff_flat_stalkMap, IdealSheafData.subschemeι_app_surjective, preimage_zeroLocus, AlgebraicGeometry.isIso_iff_stalk_iso, instIsOverMapStalkMapOverInferInstanceOverClass, AffineZariskiSite.cocone_ι_app, AlgebraicGeometry.IsAffineOpen.SpecMap_appLE_fromSpec_assoc, forgetToLocallyRingedSpace_obj, Hom.smoothLocus_eq_top, AlgebraicGeometry.Spec.map_base, compactSpace_of_isLimit, IdealSheafData.coe_support_inter, AlgebraicGeometry.compactSpace_of_universallyClosed, AlgebraicGeometry.IsAffineOpen.fromSpec_top, Hom.support_ker, AlgebraicGeometry.SpecToEquivOfLocalRing_symm_apply, AlgebraicGeometry.exists_mem_of_isClosed_of_nonempty, Hom.homeomorph_apply, Hom.stalkMap_congr, Hom.isIso_toPshHom, IdealSheafData.support_top, map_basicOpen_map, AlgebraicGeometry.affineOpensRestrict_symm_apply_coe, Hom.comp_appTop, AlgebraicGeometry.instNoetherianSpaceCarrierCarrierCommRingCatSpecOfIsNoetherianRingCarrier, AlgebraicGeometry.IsOpenImmersion.opensRange_pullback_snd_of_left, AlgebraicGeometry.IsAffineOpen.Spec_map_appLE_fromSpec, AlgebraicGeometry.IsIntegralHom.hasAffineProperty, Hom.preimage_mono, AlgebraicGeometry.instJacobsonSpaceCarrierCarrierCommRingCatSpecOfIsJacobsonRingCarrier, germ_stalkClosedPointTo_Spec, AlgebraicGeometry.exists_map_eq_top, basicOpen_zero, AlgebraicGeometry.IsAffineOpen.isLocalization_stalk', Hom.surjective, residue_residueFieldCongr_assoc, IdealSheafData.ideal_ofIdeals_le, AlgebraicGeometry.LocallyOfFinitePresentation.finitePresentation_of_affine_subset, Hom.appLE_comp_appLE_assoc, IsLocallyDirected.exists_of_pullback_V_V, SpecMap_presheaf_map_eqToHom, AlgebraicGeometry.IsAffineOpen.fromSpec_image_basicOpen, AlgebraicGeometry.instHasAffinePropertyIsomorphismsSchemeAndIsAffineIsIsoCommRingCatAppTop, AlgebraicGeometry.pullbackRestrictIsoRestrict_hom_morphismRestrict, AlgebraicGeometry.IsAffineOpen.basicOpen_union_eq_self_iff, AlgebraicGeometry.isFinite_iff, AlgebraicGeometry.etale_iff, restrict_carrier, AlgebraicGeometry.spread_out_unique_of_isGermInjective, germ_residue_assoc, AlgebraicGeometry.IsAffineOpen.Spec_map_appLE_fromSpec_assoc, AffineZariskiSite.PreservesLocalization.opensRange_map, AlgebraicGeometry.isAffineOpen_bot, IsLocallyDirected.V_self, Opens.ι_image_le, Hom.stalkMap_congr_assoc, AlgebraicGeometry.mono_pushoutSection_of_isCompact_of_flat_left, AlgebraicGeometry.IsOpenImmersion.opensRange_pullback_fst_of_right, AlgebraicGeometry.exists_appTop_map_eq_zero_of_isLimit, mem_basicOpen'', AlgebraicGeometry.stalkClosedPointIso_inv, AffineZariskiSite.cocone_pt, AlgebraicGeometry.IsZariskiLocalAtSource.iff_exists_resLE, AlgebraicGeometry.isEmpty_pullback_sigmaι_of_ne, stalkMap_inv_hom_assoc, Cover.RelativeGluingData.preimage_toBase_eq_range_ι, AlgebraicGeometry.tilde.toOpen_map_app_assoc, AlgebraicGeometry.locallyQuasiFinite_iff_finite_preimage_singleton, AlgebraicGeometry.IsClosedImmersion.SpecMap_residue, AlgebraicGeometry.exists_app_map_eq_zero_of_isLimit, AlgebraicGeometry.basicOpen_eq_of_affine, Opens.ι_appLE, ΓSpecIso_naturality_assoc, AlgebraicGeometry.IsAffineOpen.fromSpec_toSpecΓ_assoc, Hom.comp_image, AlgebraicGeometry.IsOpenImmersion.range_pullbackFst, AlgebraicGeometry.IsAffineOpen.fromSpec_image_zeroLocus, appLE_comp_appLE, Cover.instEpiTopCatBaseCommRingCatFromGlued, IdealSheafData.mem_supportSet_iff_of_mem, affineBasisCover_map_range, Hom.resLE_comp_resLE, AlgebraicGeometry.IsAffineOpen.iInf, Hom.resLE_appLE, residue_descResidueField_assoc, AlgebraicGeometry.ΓSpec_adjunction_homEquiv_eq, AlgebraicGeometry.Flat.isQuotientMap_of_surjective, Opens.ι_app, Hom.QuasiFiniteAt.isClopen_singleton_asFiber, Modules.pushforwardCongr_inv_app_app, IdealSheafData.supportSet_inter, AlgebraicGeometry.instIsPreimmersionAsFiberHom, AlgebraicGeometry.locallyOfFinitePresentation_iff, AlgebraicGeometry.exists_basicOpen_le_affine_inter, Hom.isoImage_hom_ι, AlgebraicGeometry.IsAffineOpen.isOpenImmersion_fromSpec, AlgebraicGeometry.spread_out_unique_of_isGermInjective', Hom.isCompact_preimage, AlgebraicGeometry.quasiCompact_iff_spectral, iso_hom_base_inv_base_apply, AlgebraicGeometry.instUniversallyClosedMorphismRestrict, toIso_inv_ι, AlgebraicGeometry.quasiCompact_iff_isSpectralMap, Hom.instIsIsoCommRingCatApp, residueFieldCongr_trans_hom, Hom.opensRange_comp_of_isIso, Hom.inv_image, germ_residue, AlgebraicGeometry.IsPreimmersion.instMorphismRestrict, Hom.opensRange_comp, Hom.ι_fromNormalization, Hom.comp_base, AlgebraicGeometry.exists_finite_imageι_comp_morphismRestrict_of_finite_image_preimage, AlgebraicGeometry.genericPoint_eq_bot_of_affine, AlgebraicGeometry.IsAffineOpen.isCompactOpenCovered, AlgebraicGeometry.instIsQuasicoherentOpensCarrierCarrierCommRingCatSpecTilde, AlgebraicGeometry.IsOpenImmersion.map_ΓIso_hom, Hom.map_appLE, IdealSheafData.coe_support_vanishingIdeal, Γevaluation_naturality, AlgebraicGeometry.IsSeparated.instMorphismRestrict, PartialMap.dense_domain, AlgebraicGeometry.basicOpen_eq_bot_iff, Modules.instReflectsIsomorphismsPresheafAbCarrierCommRingCatToPresheaf, AlgebraicGeometry.Proj.localRingHom_comp_stalkIso_apply, IdealSheafData.ker_glueDataObjι_appTop, AlgebraicGeometry.FormallyUnramified.formallyUnramified_appLE, AlgebraicGeometry.Spec.germ_stalkMapIso_hom, AlgebraicGeometry.Proj.homOfLE_toBasicOpenOfGlobalSections_ι, Hom.appIso_inv_app_apply, AlgebraicGeometry.HasRingHomProperty.stalkMap_of_respectsIso, restrictFunctor_obj_left, AlgebraicGeometry.ExistsHomHomCompEqCompAux.range_g_subset, AlgebraicGeometry.LocallyOfFiniteType.jacobsonSpace, AlgebraicGeometry.Spec.fromSpecStalk_eq, AlgebraicGeometry.quasiCompact_iff_forall_affine, AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv_over, Hom.stalkMap_id, AlgebraicGeometry.instLocallyOfFinitePresentationMorphismRestrict, IdealSheafData.ideal_biInf, isLocalHom_stalkClosedPointTo, AlgebraicGeometry.Proj.iSup_basicOpen_eq_top, AlgebraicGeometry.IsAffineOpen.fromSpec_preimage_basicOpen', basicOpen_restrict, IdealSheafData.support_bot, Hom.opensRange_of_isIso, AlgebraicGeometry.IsAffineOpen.app_basicOpen_eq_away_map, preimage_basicOpen, Opens.instIsIsoCommRingCatAppLEιTopToScheme, AlgebraicGeometry.IsAffineOpen.sup_of_disjoint, AlgebraicGeometry.IsAffineHom.isAffine_preimage, AlgebraicGeometry.IsAffineOpen.ΓSpecIso_hom_fromSpec_app, AlgebraicGeometry.SpecToEquivOfLocalRing_eq_iff, exists_le_and_germ_injective, Hom.flat_appLE, restrictFunctor_map_left, Hom.SpecMap_residueFieldMap_fromSpecResidueField, affineOpenCover_X, comp_app_assoc, AlgebraicGeometry.IsAffineOpen.isQuasiSeparated, Pullback.Triplet.ofPoint_y, AlgebraicGeometry.instQuasiCompactToSpecΓOfCompactSpaceCarrierCarrierCommRingCat, AlgebraicGeometry.IsAffineOpen.SpecMap_appLE_fromSpec, AlgebraicGeometry.morphismRestrict_app, AlgebraicGeometry.AffineSpace.comp_homOfVector_assoc, Hom.appLE_comp_appLE, PartialMap.le_domain_toRationalMap, Modules.instIsRightAdjointPresheafOfModulesToPresheafOfModules, Opens.ι_image_top, Hom.toImage_app, range_fromSpecResidueField, Modules.restrictFunctorId_hom_app_app, SpecMap_stalkMap_fromSpecStalk, AlgebraicGeometry.exists_lift_of_germInjective_aux, AlgebraicGeometry.isIso_pushoutSection_of_isQuasiSeparated_of_flat_right, zeroLocus_isClosed, AffineZariskiSite.instIsLocallyFullOpensToOpensFunctorGrothendieckTopologyCarrierCarrierCommRingCat, AlgebraicGeometry.IsAffineOpen.isoSpec_hom, Modules.Hom.app_smul, AlgebraicGeometry.isPullback_opens_inf_le, Γ_obj_op, Hom.stalkMap_inv_hom_apply, Hom.mem_quasiFiniteLocus, AlgebraicGeometry.quasiCompactCover_iff, image_zeroLocus, Hom.appLE_map, basicOpen_eq_bot_iff_forall_evaluation_eq_zero, AlgebraicGeometry.HasRingHomProperty.iff_of_source_openCover, Hom.isoImage_hom_ι_assoc, isBasis_affineOpens, Hom.fiberToSpecResidueField_apply, inv_base_hom_base_assoc, Opens.topIso_hom, Hom.stalkMap_hom_inv_assoc, IdealSheafData.support_sup, ofRestrict_app, AlgebraicGeometry.spread_out_of_isGermInjective, IsLocallyDirected.homOfLE_tAux_assoc, Pullback.Triplet.ofPoint_SpecTensorTo, OpenCover.restrict_X, zeroLocus_univ, JointlySurjective.exists_eq, Hom.image_le_image_iff, Hom.comp_appTop_assoc, AlgebraicGeometry.IsAffineOpen.fromSpec_app_of_le, AlgebraicGeometry.AffineSpace.map_appTop_coord, toSpecΓ_preimage_basicOpen, residueFieldCongr_trans_hom_assoc, Hom.isIrreducible_preimage, Hom.app_appIso_inv, AlgebraicGeometry.isIso_fromTildeΓ_iff, residueFieldCongr_inv, isEmpty_pullback, coe_homeoOfIso, Hom.normalizationObjIso_hom_val, openCoverBasicOpenTop_f, zero_of_zero_cover, AlgebraicGeometry.pointEquivClosedPoint_symm_apply_coe, Hom.appLE_eq_app, RationalMap.mem_domain, IdealSheafData.support_sSup, AlgebraicGeometry.instHasAffinePropertyQuasiCompactCompactSpaceCarrierCarrierCommRingCat, AlgebraicGeometry.IsOpenImmersion.app_eq_invApp_app_of_comp_eq_aux, AlgebraicGeometry.isField_of_isIntegral_of_subsingleton, AlgebraicGeometry.spec_punit_isEmpty, AlgebraicGeometry.isClosedMap_iff_specializingMap, PartialMap.exists_restrict_isOver, AlgebraicGeometry.instIsNoetherianRingCarrierStalkCommRingCatPresheafOfIsLocallyNoetherian, AlgebraicGeometry.exists_of_res_eq_of_qcqs, AlgebraicGeometry.isReduced_stalk_of_isReduced, image_preimage_eq_of_isPullback, AlgebraicGeometry.ProjectiveSpectrum.Proj.toOpen_toSpec_val_c_app, Hom.instIsLocalHomCarrierStalkCommRingCatPresheafCoeContinuousMapCarrierCarrierHomTopCatBaseRingHomHomStalkMap, GlueData.ι_jointly_surjective, AffineZariskiSite.PreservesLocalization.colimitDesc_preimage, IdealSheafData.vanishingIdeal_iSup, AlgebraicGeometry.isLocalization_basicOpen_of_qcqs, IdealSheafData.strictMono_ideal, codisjoint_zeroLocus, Opens.eq_presheaf_map_eqToHom, PartialMap.restrict_id_hom, stalkClosedPointTo_comp, quasiSeparatedSpace_of_isOpenCover, Opens.toSpecΓ_SpecMap_appLE, AlgebraicGeometry.IsAffineOpen.fromSpec_toSpecΓ, AlgebraicGeometry.map_injective_of_isIntegral, AlgebraicGeometry.exists_eq_pow_mul_of_isCompact_of_isQuasiSeparated, AlgebraicGeometry.AffineSpace.homOfVector_appTop_coord, Spec_map_residue_apply, AlgebraicGeometry.sourceAffineLocally_morphismRestrict, IsQuasiAffine.isBasis_basicOpen, AlgebraicGeometry.initial_isEmpty, residueFieldMap_comp, AlgebraicGeometry.Etale.instMorphismRestrict, isoSpec_image_zeroLocus, Hom.fiberι_fiberHomeo_symm, comp_coeBase, IdealSheafData.ideal_map_of_isAffineHom, Spec_stalkClosedPointTo_fromSpecStalk, AlgebraicGeometry.isCompact_and_isOpen_iff_finite_and_eq_biUnion_basicOpen, Hom.le_resLE_preimage_iff, AlgebraicGeometry.IsAffineOpen.toSpecΓ_isoSpec_inv_assoc, Modules.pushforward_map_app, mem_toGrothendieck_smallPretopology, IsJointlySurjectivePreserving.exists_preimage_fst_triplet_of_prop, Hom.stalkMap_inv_hom, AlgebraicGeometry.coe_opensRestrict_apply_coe, AlgebraicGeometry.Proj.localRingHom_comp_stalkIso, AffineZariskiSite.presieveOfSections_eq_ofArrows, Hom.inv_app, localRingHom_comp_stalkIso, restrictFunctor_obj_hom, Hom.isQuasiSeparated_preimage, AlgebraicGeometry.finite_irreducibleComponents_of_isNoetherian, Hom.app_injective, Modules.inv_app, AlgebraicGeometry.GeometricallyIrreducible.irreducibleSpace, Hom.Spec_map_residueFieldMap_fromSpecResidueField, Opens.toSpecΓ_preimage_zeroLocus, Cover.fromGlued_open_map, exists_isOpenCover_and_isAffine, AlgebraicGeometry.coprodMk_inr, instEpiCommRingCatResidue, Hom.isProperMap, isoOfEq_hom, AlgebraicGeometry.QuasiCompactCover.exists_isAffineOpen_of_isCompact, stalkMap_germ, le_iff_specializes, Hom.asFiberHom_fiberToSpecResidueField_assoc, AlgebraicGeometry.quasiSeparatedSpace_iff_quasiCompact_prod_lift, Spec.residue_residueFieldIso_hom, IdealSheafData.vanishingIdeal_sSup, Pullback.exists_preimage_pullback, AffineZariskiSite.mem_grothendieckTopology_iff_sectionsOfPresieve, AlgebraicGeometry.IsAffineOpen.isoSpec_inv_ι, AlgebraicGeometry.Proj.fromOfGlobalSections_preimage_basicOpen, AlgebraicGeometry.IsAffineOpen.isoSpec_hom_appTop, IdealSheafData.map_ideal_basicOpen, AffineZariskiSite.toOpensFunctor_obj, localRingHom_comp_stalkIso_apply, AlgebraicGeometry.instIsIsoModulesSpecOfCarrierFromTildeΓFreeOpensCarrierCarrierCommRingCat, AlgebraicGeometry.IsOpenImmersion.comp_lift, Hom.isOpen_smoothLocus, mem_zeroLocus_iff, comp_appLE_assoc, Hom.fromNormalization_app, AlgebraicGeometry.Proj.awayToSection_comp_appLE_assoc, AlgebraicGeometry.Spec_stalkClosedPointIso, AlgebraicGeometry.spread_out_of_isGermInjective', stalkSpecializes_stalkMap_assoc, Opens.ι_base_apply, Hom.eqToHom_app, Hom.isDiscrete_preimage, AlgebraicGeometry.instIsAffineHomιBasicOpen, id_app, AlgebraicGeometry.flat_iff, homOfLE_apply, zeroLocus_map_of_eq, evaluation_ne_zero_iff_mem_basicOpen, AlgebraicGeometry.IsAffineOpen.toSpecΓ_fromSpec, residueFieldCongr_symm, PartialMap.fromSpecStalkOfMem_toPartialMap, AlgebraicGeometry.IsAffineOpen.fromSpec_primeIdealOf, exists_preimage_of_isPullback, homOfLE_rfl, fromSpecResidueField_apply, AlgebraicGeometry.IsOpenImmersion.lift_app, AffineZariskiSite.instIsCoverDenseOpensToOpensFunctorGrothendieckTopologyCarrierCarrierCommRingCat, AlgebraicGeometry.IsAffineOpen.isoSpec_inv_ι_assoc, AlgebraicGeometry.HasRingHomProperty.iff_of_isAffine, Hom.coe_opensRange, AlgebraicGeometry.quasiSeparatedSpace_iff_affine, IdealSheafData.le_def, stalkMap_congr, AffineZariskiSite.restrictIsoSpec_inv_app, AlgebraicGeometry.isClosedImmersion_diagonal_restrict_diagonalCoverDiagonalRange, AlgebraicGeometry.isIso_pushoutSection_of_isAffineOpen, Hom.stalkMap_congr_hom, AlgebraicGeometry.isCompact_basicOpen, inv_base_hom_base, IdealSheafData.vanishingIdeal_bot, AlgebraicGeometry.tilde.instIsLocalizedModuleCarrierCarrierObjOppositeOpensCarrierCarrierCommRingCatSpecModuleCatPresheafModulesSheafModulesSpecToSheafOpBasicOpenPowersHomToOpen, isNilpotent_iff_basicOpen_eq_bot_of_isCompact, AlgebraicGeometry.quasiSeparated_over_affine_iff, IdealSheafData.subschemeι_apply, IdealSheafData.supportSet_eq_iInter_zeroLocus, AlgebraicGeometry.IsAffineOpen.appLE_eq_away_map, Hom.appIso_inv_appLE, Modules.map_smul, mem_basicOpen, AlgebraicGeometry.IsOpenImmersion.affineOpensEquiv_symm_apply_coe, affineBasicOpen_le, Spec_stalkClosedPointTo_fromSpecStalk_assoc, AlgebraicGeometry.SpecMap_ΓSpecIso_inv_toSpecΓ_assoc, AlgebraicGeometry.IsOpenImmersion.map_ΓIso_inv_apply, AlgebraicGeometry.IsAffineOpen.preimage, Opens.isoOfLE_inv_ι_assoc, AlgebraicGeometry.Spec.map_app, AlgebraicGeometry.exists_lift_of_germInjective, Spec_fromSpecStalk, Cover.fromGlued_injective, AlgebraicGeometry.IsAffineOpen.ideal_ext_iff, AlgebraicGeometry.exists_etale_isCompl_of_quasiFiniteAt, AlgebraicGeometry.Spec.map_base_apply, AlgebraicGeometry.morphismRestrict_id, Hom.stalkMap_surjective, AlgebraicGeometry.tilde.toOpen_map_app, AlgebraicGeometry.IsAffineOpen.primeIdealOf_genericPoint, Hom.id_image, AlgebraicGeometry.IsAffineOpen.mem_ideal_iff, toSpecΓ_base, AlgebraicGeometry.Smooth.iff_forall_exists_isStandardSmooth
zeroLocus 📖	CompOp	47 math math: zeroLocus_iInf, zeroLocus_empty_eq_univ, zeroLocus_eq_univ_iff_subset_nilradical, zeroLocus_mul, isoSpec_inv_preimage_zeroLocus, zeroLocus_biInf, zeroLocus_iUnion, zeroLocus_singleton, IdealSheafData.mem_support_iff_of_mem, isoSpec_inv_image_zeroLocus, IdealSheafData.range_glueDataObjι_ι, zeroLocus_setMul, toSpecΓ_preimage_zeroLocus, zeroLocus_mono, zeroLocus_def, IdealSheafData.mem_supportSet_iff, IdealSheafData.mem_support_iff, IdealSheafData.coe_support_ofIdealTop, zeroLocus_iInf_of_nonempty, zeroLocus_radical, zeroLocus_map, AlgebraicGeometry.IsAffineOpen.fromSpec_preimage_zeroLocus, zeroLocus_inf, zeroLocus_eq_univ_iff_subset_nilradical_of_isCompact, AlgebraicGeometry.Spec_zeroLocus_eq_zeroLocus, eq_zeroLocus_of_isClosed_of_isAffine, toSpecΓ_image_zeroLocus, AlgebraicGeometry.Spec_zeroLocus, IdealSheafData.coe_support_eq_eq_iInter_zeroLocus, zeroLocus_biInf_of_nonempty, IdealSheafData.supportSet_subset_zeroLocus, IdealSheafData.zeroLocus_inter_subset_supportSet, zeroLocus_span, preimage_zeroLocus, IdealSheafData.coe_support_inter, AlgebraicGeometry.IsAffineOpen.fromSpec_image_zeroLocus, IdealSheafData.mem_supportSet_iff_of_mem, IdealSheafData.supportSet_inter, zeroLocus_isClosed, image_zeroLocus, zeroLocus_univ, codisjoint_zeroLocus, isoSpec_image_zeroLocus, Opens.toSpecΓ_preimage_zeroLocus, mem_zeroLocus_iff, zeroLocus_map_of_eq, IdealSheafData.supportSet_eq_iInter_zeroLocus
Γ 📖	CompOp	24 math math: AlgebraicGeometry.ΓSpec.isIso_adjunction_counit, restrictFunctorΓ_inv_app, AlgebraicGeometry.preservesColimit_Γ, AlgebraicGeometry.instPreservesLimitSchemeOppositeCommRingCatRightOpΓOfIsAffineHomMapOfCompactSpaceOfQuasiSeparatedSpaceCarrierCarrierObj, SpecΓIdentity_hom_app, Γ_obj, AlgebraicGeometry.ΓSpec.adjunction_counit_app', AlgebraicGeometry.ΓSpec.adjunction_counit_app, Γ_def, SpecΓIdentity_app, AlgebraicGeometry.ΓSpec.adjunction_unit_app, restrictFunctorΓ_hom_app, Γ_map_op, AlgebraicGeometry.instIsIsoSchemeAppUnitOppositeCommRingCatAdjunctionOfIsAffine, AlgebraicGeometry.instIsRightAdjointCommRingCatOppositeSchemeΓ, Γ_map, AlgebraicGeometry.ΓSpec.adjunction_homEquiv_apply, AlgebraicGeometry.ΓSpecIso_inv_ΓSpec_adjunction_homEquiv, SpecΓIdentity_inv_app, AlgebraicGeometry.preservesLimit_rightOp_Γ, AlgebraicGeometry.ΓSpec.adjunction_homEquiv_symm_apply, AlgebraicGeometry.ΓSpec_adjunction_homEquiv_eq, Γ_obj_op, AlgebraicGeometry.nonempty_isColimit_Γ_mapCocone
ΓSpecIso 📖	CompOp	64 math math: ΓSpecIso_inv_naturality_assoc, AlgebraicGeometry.IsAffineOpen.isoSpec_inv_appTop, germ_stalkClosedPointTo, AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv, AlgebraicGeometry.AffineSpace.SpecIso_hom_appTop, ideal_ker_le_ker_ΓSpecIso_inv_comp, SpecΓIdentity_hom_app, AlgebraicGeometry.ProjectiveSpectrum.Proj.toOpen_toSpec_val_c_app_assoc, AlgebraicGeometry.Proj.basicOpenToSpec_app_top, AlgebraicGeometry.SpecMap_ΓSpecIso_hom, AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv_appTop_coord, ΓSpecIso_inv, AlgebraicGeometry.SpecMap_ΓSpecIso_inv_toSpecΓ, fromSpecStalk_appTop, Opens.toSpecΓ_appTop_assoc, AlgebraicGeometry.IsAffineOpen.algebraMap_Spec_obj, AlgebraicGeometry.ΓSpec.adjunction_counit_app, Opens.toSpecΓ_appTop, AlgebraicGeometry.ProjectiveSpectrum.Proj.toStalk_stalkMap_toSpec_assoc, AlgebraicGeometry.ProjectiveSpectrum.Proj.toStalk_stalkMap_toSpec, AlgebraicGeometry.ΓSpecIso_obj_hom, SpecΓIdentity_app, AlgebraicGeometry.Spec.germ_stalkMapIso_hom_assoc, AlgebraicGeometry.AffineSpace.SpecIso_inv_appTop_coord, AlgebraicGeometry.germ_stalkClosedPointIso_hom_assoc, AlgebraicGeometry.IsAffineOpen.fromSpec_app_self, isoSpec_Spec_hom, ΓSpecIso_naturality, AlgebraicGeometry.germ_stalkClosedPointIso_hom, toSpecΓ_appTop, AlgebraicGeometry.Spec_zeroLocus_eq_zeroLocus, isoSpec_Spec, Spec.algebraMap_residueFieldIso_inv, AlgebraicGeometry.toSpecΓ_SpecMap_ΓSpecIso_inv_assoc, AlgebraicGeometry.IsAffineOpen.fromSpec_app_self_apply, AlgebraicGeometry.ΓSpecIso_hom_stalkClosedPointIso_inv, AlgebraicGeometry.AffineSpace.toSpecMvPolyIntEquiv_apply, AlgebraicGeometry.Spec.algebraMap_stalkIso_inv, toOpen_eq, AlgebraicGeometry.ΓSpecIso_inv_ΓSpec_adjunction_homEquiv, AlgebraicGeometry.toSpecΓ_SpecMap_ΓSpecIso_inv, AlgebraicGeometry.Spec_zeroLocus, ΓSpecIso_inv_naturality, SpecΓIdentity_inv_app, AlgebraicGeometry.basicOpen_eq_of_affine', AlgebraicGeometry.AffineSpace.isoOfIsAffine_hom_appTop, isoSpec_Spec_inv, germ_stalkClosedPointTo_assoc, AlgebraicGeometry.Spec.algebraMap_stalkIso_inv_assoc, fromSpecStalk_app, Spec.algebraMap_residueFieldIso_inv_assoc, germ_stalkClosedPointTo_Spec, AlgebraicGeometry.basicOpen_eq_of_affine, ΓSpecIso_naturality_assoc, AlgebraicGeometry.ΓSpec_adjunction_homEquiv_eq, AlgebraicGeometry.Spec.germ_stalkMapIso_hom, AlgebraicGeometry.Spec.fromSpecStalk_eq, AlgebraicGeometry.IsAffineOpen.fromSpec_preimage_basicOpen', AlgebraicGeometry.IsAffineOpen.ΓSpecIso_hom_fromSpec_app, AlgebraicGeometry.IsAffineOpen.fromSpec_app_of_le, AlgebraicGeometry.ProjectiveSpectrum.Proj.toOpen_toSpec_val_c_app, AlgebraicGeometry.IsAffineOpen.isoSpec_hom_appTop, AlgebraicGeometry.SpecMap_ΓSpecIso_inv_toSpecΓ_assoc, Spec_fromSpecStalk

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
SpecMap_presheaf_map_eqToHom 📖	mathematical	—	Hom.app AlgebraicGeometry.Spec CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op Opens AlgebraicGeometry.Spec.map CategoryTheory.Functor.map Quiver.Hom.op CategoryTheory.CategoryStruct.toQuiver CategoryTheory.Category.toCategoryStruct AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier CategoryTheory.eqToHom TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom'	—	CategoryTheory.Functor.map_id CategoryTheory.op_id Hom.congr_app AlgebraicGeometry.Spec.map_id TopologicalSpace.Opens.map_id_obj CategoryTheory.eqToHom_op CategoryTheory.eqToHom_map CategoryTheory.eqToHom_naturality CategoryTheory.Category.comp_id
Spec_map 📖	mathematical	—	CategoryTheory.Functor.map Opposite CommRingCat CategoryTheory.Category.opposite CommRingCat.instCategory AlgebraicGeometry.Scheme instCategory Spec AlgebraicGeometry.Spec.map Opposite.unop Quiver.Hom.unop CategoryTheory.CategoryStruct.toQuiver CategoryTheory.Category.toCategoryStruct	—	—
Spec_map_presheaf_map_eqToHom 📖	mathematical	—	Hom.app AlgebraicGeometry.Spec CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op Opens AlgebraicGeometry.Spec.map CategoryTheory.Functor.map Quiver.Hom.op CategoryTheory.CategoryStruct.toQuiver CategoryTheory.Category.toCategoryStruct AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier CategoryTheory.eqToHom TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom'	—	SpecMap_presheaf_map_eqToHom
Spec_obj 📖	mathematical	—	CategoryTheory.Functor.obj Opposite CommRingCat CategoryTheory.Category.opposite CommRingCat.instCategory AlgebraicGeometry.Scheme instCategory Spec AlgebraicGeometry.Spec Opposite.unop	—	—
SpecΓIdentity_app 📖	mathematical	—	CategoryTheory.Iso.app CommRingCat CommRingCat.instCategory CategoryTheory.Functor.comp Opposite AlgebraicGeometry.Scheme CategoryTheory.Category.opposite instCategory CategoryTheory.Functor.rightOp Spec Γ CategoryTheory.Functor.id SpecΓIdentity ΓSpecIso	—	—
SpecΓIdentity_hom_app 📖	mathematical	—	CategoryTheory.NatTrans.app CommRingCat CommRingCat.instCategory CategoryTheory.Functor.comp Opposite AlgebraicGeometry.Scheme CategoryTheory.Category.opposite instCategory CategoryTheory.Functor.rightOp Spec Γ CategoryTheory.Functor.id CategoryTheory.Iso.hom CategoryTheory.Functor CategoryTheory.Functor.category SpecΓIdentity CategoryTheory.Functor.obj TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.Spec TopCat.str Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op Opens Top.top OrderTop.toTop Preorder.toLE SemilatticeSup.toPartialOrder Lattice.toSemilatticeSup CompleteLattice.toLattice TopologicalSpace.Opens.instCompleteLattice AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier BoundedOrder.toOrderTop CompleteLattice.toBoundedOrder ΓSpecIso	—	—
SpecΓIdentity_inv_app 📖	mathematical	—	CategoryTheory.NatTrans.app CommRingCat CommRingCat.instCategory CategoryTheory.Functor.id CategoryTheory.Functor.comp Opposite AlgebraicGeometry.Scheme CategoryTheory.Category.opposite instCategory CategoryTheory.Functor.rightOp Spec Γ CategoryTheory.Iso.inv CategoryTheory.Functor CategoryTheory.Functor.category SpecΓIdentity CategoryTheory.Functor.obj TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.Spec TopCat.str Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op Opens Top.top OrderTop.toTop Preorder.toLE SemilatticeSup.toPartialOrder Lattice.toSemilatticeSup CompleteLattice.toLattice TopologicalSpace.Opens.instCompleteLattice AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier BoundedOrder.toOrderTop CompleteLattice.toBoundedOrder ΓSpecIso	—	—
appLE_comp_appLE 📖	mathematical	Opens Preorder.toLE PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier CategoryTheory.Functor.obj Preorder.smallCategory TopologicalSpace.Opens TopCat.str TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom'	CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op Opens Hom.appLE AlgebraicGeometry.Scheme instCategory LE.le.trans AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom' Quiver.Hom.le CategoryTheory.Functor.map CategoryTheory.homOfLE	—	Hom.appLE_comp_appLE
app_eq 📖	mathematical	—	Hom.app CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op Opens AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom' CategoryTheory.Functor.map Quiver.Hom.op CategoryTheory.CategoryStruct.toQuiver CategoryTheory.eqToHom	—	Hom.app_eq
basicOpen_add_le 📖	mathematical	—	Opens Preorder.toLE PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier basicOpen CommRingCat.carrier CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op Distrib.toAdd NonUnitalNonAssocSemiring.toDistrib NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing CommRing.toNonUnitalCommRing CommRingCat.commRing SemilatticeSup.toMax Lattice.toSemilatticeSup ConditionallyCompleteLattice.toLattice CompleteLattice.toConditionallyCompleteLattice TopologicalSpace.Opens.instCompleteLattice	—	basicOpen_le CommRingCat.Colimits.hasColimits_commRingCat mem_basicOpen IsLocalRing.isUnit_or_isUnit_of_isUnit_add AlgebraicGeometry.LocallyRingedSpace.instIsLocalRingCarrierStalkCommRingCatPresheaf map_add AddMonoidHomClass.toAddHomClass RingHomClass.toAddMonoidHomClass RingHom.instRingHomClass
basicOpen_appLE 📖	mathematical	Opens Preorder.toLE PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier CategoryTheory.Functor.obj Preorder.smallCategory TopologicalSpace.Opens TopCat.str TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom'	basicOpen DFunLike.coe CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op Opens CommRingCat.carrier CategoryTheory.ConcreteCategory.hom RingHom Semiring.toNonAssocSemiring CommSemiring.toSemiring CommRing.toCommSemiring CommRingCat.commRing RingHom.instFunLike CommRingCat.instConcreteCategoryRingHomCarrier Hom.appLE SemilatticeInf.toMin Lattice.toSemilatticeInf ConditionallyCompleteLattice.toLattice CompleteLattice.toConditionallyCompleteLattice TopologicalSpace.Opens.instCompleteLattice AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom'	—	CommRingCat.comp_apply preimage_basicOpen basicOpen_res
basicOpen_le 📖	mathematical	—	Opens Preorder.toLE PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier basicOpen	—	AlgebraicGeometry.RingedSpace.basicOpen_le
basicOpen_mul 📖	mathematical	—	basicOpen CommRingCat.carrier CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op Opens Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing CommRing.toNonUnitalCommRing CommRingCat.commRing SemilatticeInf.toMin Lattice.toSemilatticeInf ConditionallyCompleteLattice.toLattice CompleteLattice.toConditionallyCompleteLattice TopologicalSpace.Opens.instCompleteLattice AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier	—	AlgebraicGeometry.RingedSpace.basicOpen_mul
basicOpen_of_isUnit 📖	mathematical	IsUnit CommRingCat.carrier CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op Opens MonoidWithZero.toMonoid Semiring.toMonoidWithZero CommSemiring.toSemiring CommRing.toCommSemiring CommRingCat.commRing	basicOpen	—	AlgebraicGeometry.RingedSpace.basicOpen_of_isUnit
basicOpen_one 📖	mathematical	—	basicOpen CommRingCat.carrier CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op Opens AddMonoidWithOne.toOne AddGroupWithOne.toAddMonoidWithOne Ring.toAddGroupWithOne CommRing.toRing CommRingCat.commRing	—	basicOpen_of_isUnit isUnit_one
basicOpen_pow 📖	mathematical	—	basicOpen CommRingCat.carrier CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op Opens Monoid.toPow MonoidWithZero.toMonoid Semiring.toMonoidWithZero CommSemiring.toSemiring CommRing.toCommSemiring CommRingCat.commRing	—	AlgebraicGeometry.RingedSpace.basicOpen_pow
basicOpen_res 📖	mathematical	—	basicOpen DFunLike.coe CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op Opens CommRingCat.carrier CategoryTheory.ConcreteCategory.hom RingHom Semiring.toNonAssocSemiring CommSemiring.toSemiring CommRing.toCommSemiring CommRingCat.commRing RingHom.instFunLike CommRingCat.instConcreteCategoryRingHomCarrier CategoryTheory.Functor.map SemilatticeInf.toMin Lattice.toSemilatticeInf ConditionallyCompleteLattice.toLattice CompleteLattice.toConditionallyCompleteLattice TopologicalSpace.Opens.instCompleteLattice AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier	—	AlgebraicGeometry.RingedSpace.basicOpen_res
basicOpen_res_eq 📖	mathematical	—	basicOpen DFunLike.coe CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op Opens CommRingCat.carrier CategoryTheory.ConcreteCategory.hom RingHom Semiring.toNonAssocSemiring CommSemiring.toSemiring CommRing.toCommSemiring CommRingCat.commRing RingHom.instFunLike CommRingCat.instConcreteCategoryRingHomCarrier CategoryTheory.Functor.map	—	AlgebraicGeometry.RingedSpace.basicOpen_res_eq
basicOpen_restrict 📖	mathematical	—	Opens Preorder.toLE PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier basicOpen TopCat.Presheaf.restrict RingHom CommRingCat.carrier Semiring.toNonAssocSemiring CommSemiring.toSemiring CommRing.toCommSemiring CommRingCat.commRing RingHom.instFunLike CommRingCat.instConcreteCategoryRingHomCarrier AlgebraicGeometry.PresheafedSpace.presheaf	—	Eq.trans_le basicOpen_res inf_le_right
basicOpen_zero 📖	mathematical	—	basicOpen CommRingCat.carrier CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op Opens MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing CommRing.toNonUnitalCommRing CommRingCat.commRing Bot.bot OrderBot.toBot Preorder.toLE SemilatticeSup.toPartialOrder Lattice.toSemilatticeSup GeneralizedHeytingAlgebra.toLattice HeytingAlgebra.toGeneralizedHeytingAlgebra Order.Frame.toHeytingAlgebra TopologicalSpace.Opens.instFrame AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier HeytingAlgebra.toOrderBot	—	AlgebraicGeometry.LocallyRingedSpace.basicOpen_zero
codisjoint_zeroLocus 📖	mathematical	—	Codisjoint Set TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace OmegaCompletePartialOrder.toPartialOrder CompleteLattice.instOmegaCompletePartialOrder CompleteBooleanAlgebra.toCompleteLattice CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra Set.instCompleteAtomicBooleanAlgebra Set.instOrderTop zeroLocus SetLike.coe Opens TopologicalSpace.Opens.instSetLike AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier	—	basicOpen_le
coe_homeoOfIso 📖	mathematical	—	DFunLike.coe Homeomorph TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier EquivLike.toFunLike Homeomorph.instEquivLike homeoOfIso CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap TopCat.str ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom' CategoryTheory.Iso.hom AlgebraicGeometry.Scheme instCategory	—	—
coe_homeoOfIso_symm 📖	mathematical	—	DFunLike.coe Homeomorph TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier EquivLike.toFunLike Homeomorph.instEquivLike homeoOfIso CategoryTheory.Iso.symm AlgebraicGeometry.Scheme instCategory CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap TopCat.str ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom' CategoryTheory.Iso.inv	—	—
comp_app 📖	mathematical	—	Hom.app CategoryTheory.CategoryStruct.comp AlgebraicGeometry.Scheme CategoryTheory.Category.toCategoryStruct instCategory CommRingCat CommRingCat.instCategory CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op Opens AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom'	—	Hom.comp_app
comp_appLE 📖	mathematical	Opens Preorder.toLE PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier CategoryTheory.Functor.obj Preorder.smallCategory TopologicalSpace.Opens TopCat.str TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom' CategoryTheory.CategoryStruct.comp AlgebraicGeometry.Scheme CategoryTheory.Category.toCategoryStruct instCategory	Hom.appLE CategoryTheory.CategoryStruct.comp AlgebraicGeometry.Scheme CategoryTheory.Category.toCategoryStruct instCategory CommRingCat CommRingCat.instCategory CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op Opens AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom' Hom.app	—	Hom.comp_appLE
comp_appLE_assoc 📖	mathematical	Opens Preorder.toLE PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier CategoryTheory.Functor.obj Preorder.smallCategory TopologicalSpace.Opens TopCat.str TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom' CategoryTheory.CategoryStruct.comp AlgebraicGeometry.Scheme CategoryTheory.Category.toCategoryStruct instCategory	CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op Opens Hom.appLE AlgebraicGeometry.Scheme instCategory AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom' Hom.app	—	Hom.comp_appLE_assoc
comp_appTop 📖	mathematical	—	Hom.appTop CategoryTheory.CategoryStruct.comp AlgebraicGeometry.Scheme CategoryTheory.Category.toCategoryStruct instCategory CommRingCat CommRingCat.instCategory CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op Opens Top.top OrderTop.toTop Preorder.toLE SemilatticeSup.toPartialOrder Lattice.toSemilatticeSup CompleteLattice.toLattice TopologicalSpace.Opens.instCompleteLattice AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier BoundedOrder.toOrderTop CompleteLattice.toBoundedOrder	—	Hom.comp_appTop
comp_appTop_assoc 📖	mathematical	—	CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op Opens Top.top OrderTop.toTop Preorder.toLE SemilatticeSup.toPartialOrder Lattice.toSemilatticeSup CompleteLattice.toLattice TopologicalSpace.Opens.instCompleteLattice AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier BoundedOrder.toOrderTop CompleteLattice.toBoundedOrder Hom.appTop AlgebraicGeometry.Scheme instCategory	—	Hom.comp_appTop_assoc
comp_app_assoc 📖	mathematical	—	CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op Opens AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom' AlgebraicGeometry.Scheme instCategory Hom.app	—	Hom.comp_app_assoc
comp_base 📖	mathematical	—	AlgebraicGeometry.PresheafedSpace.Hom.base CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom' CategoryTheory.CategoryStruct.comp AlgebraicGeometry.Scheme CategoryTheory.Category.toCategoryStruct instCategory TopCat TopCat.instCategory AlgebraicGeometry.PresheafedSpace.carrier	—	Hom.comp_base
comp_base_apply 📖	mathematical	—	DFunLike.coe AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace TopCat.carrier CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap TopCat.str ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom' CategoryTheory.CategoryStruct.comp AlgebraicGeometry.Scheme CategoryTheory.Category.toCategoryStruct instCategory	—	Hom.comp_apply
comp_base_assoc 📖	mathematical	—	CategoryTheory.CategoryStruct.comp TopCat CategoryTheory.Category.toCategoryStruct TopCat.instCategory AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom' AlgebraicGeometry.Scheme instCategory	—	Hom.comp_base_assoc
comp_coeBase 📖	mathematical	—	AlgebraicGeometry.PresheafedSpace.Hom.base CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom' CategoryTheory.CategoryStruct.comp AlgebraicGeometry.Scheme CategoryTheory.Category.toCategoryStruct instCategory TopCat TopCat.instCategory AlgebraicGeometry.PresheafedSpace.carrier	—	Hom.comp_base
comp_coeBase_assoc 📖	mathematical	—	CategoryTheory.CategoryStruct.comp TopCat CategoryTheory.Category.toCategoryStruct TopCat.instCategory AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom' AlgebraicGeometry.Scheme instCategory	—	Hom.comp_base_assoc
comp_toLRSHom 📖	mathematical	—	Hom.toLRSHom CategoryTheory.CategoryStruct.comp AlgebraicGeometry.Scheme CategoryTheory.Category.toCategoryStruct instCategory AlgebraicGeometry.LocallyRingedSpace AlgebraicGeometry.LocallyRingedSpace.instCategory toLocallyRingedSpace	—	Hom.comp_toLRSHom
comp_toLRSHom_assoc 📖	mathematical	—	CategoryTheory.CategoryStruct.comp AlgebraicGeometry.LocallyRingedSpace CategoryTheory.Category.toCategoryStruct AlgebraicGeometry.LocallyRingedSpace.instCategory toLocallyRingedSpace Hom.toLRSHom AlgebraicGeometry.Scheme instCategory	—	Hom.comp_toLRSHom_assoc
congr_app 📖	mathematical	—	Hom.app CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op Opens AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom' CategoryTheory.Functor.map Quiver.Hom.op CategoryTheory.CategoryStruct.toQuiver CategoryTheory.eqToHom	—	Hom.congr_app
default_asIdeal 📖	mathematical	—	PrimeSpectrum.asIdeal CommRingCat.carrier CommRingCat.of EuclideanDomain.toCommRing Field.toEuclideanDomain CommRing.toCommSemiring CommRingCat.commRing TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.Spec Unique.instInhabited instUniqueCarrierCarrierCommRingCatSpecOf Bot.bot Ideal CommSemiring.toSemiring Submodule.instBot NonUnitalNonAssocSemiring.toAddCommMonoid NonAssocSemiring.toNonUnitalNonAssocSemiring Semiring.toNonAssocSemiring Semiring.toModule	—	—
empty_carrier_carrier 📖	mathematical	—	TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace empty	—	—
empty_presheaf 📖	mathematical	—	AlgebraicGeometry.PresheafedSpace.presheaf CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace empty CategoryTheory.Functor.obj CategoryTheory.Functor Opposite TopologicalSpace.Opens TopCat.carrier TopCat.of instTopologicalSpacePEmpty TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder CategoryTheory.Functor.category CategoryTheory.Functor.const CommRingCat.of PUnit.commRing	—	—
eqToHom_c_app 📖	mathematical	—	Hom.app CategoryTheory.eqToHom AlgebraicGeometry.Scheme CategoryTheory.Category.toCategoryStruct instCategory CommRingCat CommRingCat.instCategory CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op Opens AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom'	—	Hom.eqToHom_app
forgetToLocallyRingedSpace_map 📖	mathematical	—	CategoryTheory.Functor.map AlgebraicGeometry.Scheme instCategory AlgebraicGeometry.LocallyRingedSpace AlgebraicGeometry.LocallyRingedSpace.instCategory forgetToLocallyRingedSpace Hom.toLRSHom	—	—
forgetToLocallyRingedSpace_obj 📖	mathematical	—	CategoryTheory.Functor.obj AlgebraicGeometry.Scheme instCategory AlgebraicGeometry.LocallyRingedSpace AlgebraicGeometry.LocallyRingedSpace.instCategory forgetToLocallyRingedSpace toLocallyRingedSpace	—	—
forgetToTop_comp_forget 📖	mathematical	—	CategoryTheory.Functor.comp AlgebraicGeometry.Scheme instCategory TopCat TopCat.instCategory CategoryTheory.types forgetToTop CategoryTheory.forget ContinuousMap TopCat.carrier TopCat.str ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier forget	—	—
forgetToTop_map 📖	mathematical	—	CategoryTheory.Functor.map AlgebraicGeometry.Scheme instCategory TopCat TopCat.instCategory forgetToTop AlgebraicGeometry.SheafedSpace CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.instCategory AlgebraicGeometry.SheafedSpace.forget AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace CategoryTheory.InducedCategory.homMk AlgebraicGeometry.PresheafedSpace AlgebraicGeometry.PresheafedSpace.categoryOfPresheafedSpaces AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom	—	—
forgetToTop_obj 📖	mathematical	—	CategoryTheory.Functor.obj AlgebraicGeometry.Scheme instCategory TopCat TopCat.instCategory forgetToTop AlgebraicGeometry.SheafedSpace CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.instCategory AlgebraicGeometry.SheafedSpace.forget AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace	—	—
forget_map 📖	mathematical	—	CategoryTheory.Functor.map AlgebraicGeometry.Scheme instCategory CategoryTheory.types forget DFunLike.coe AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace TopCat.carrier CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap TopCat.str ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom'	—	—
forget_obj 📖	mathematical	—	CategoryTheory.Functor.obj AlgebraicGeometry.Scheme instCategory CategoryTheory.types forget TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace	—	—
fullyFaithfulForgetToLocallyRingedSpace_preimage_toLRSHom 📖	mathematical	—	Hom.toLRSHom CategoryTheory.Functor.FullyFaithful.preimage AlgebraicGeometry.Scheme instCategory AlgebraicGeometry.LocallyRingedSpace AlgebraicGeometry.LocallyRingedSpace.instCategory forgetToLocallyRingedSpace fullyFaithfulForgetToLocallyRingedSpace	—	—
height_of_isClosed 📖	mathematical	—	Order.height TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace instPreorderCarrierCarrierCommRingCat ENat MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonAssocSemiring.toNonUnitalNonAssocSemiring Semiring.toNonAssocSemiring CommSemiring.toSemiring instCommSemiringENat	—	eq_or_ne IsClosed.not_specializes
hom_base_inv_base 📖	mathematical	—	CategoryTheory.CategoryStruct.comp TopCat CategoryTheory.Category.toCategoryStruct TopCat.instCategory AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom' CategoryTheory.Iso.hom AlgebraicGeometry.Scheme instCategory CategoryTheory.Iso.inv CategoryTheory.CategoryStruct.id	—	AlgebraicGeometry.LocallyRingedSpace.iso_hom_base_inv_base
hom_base_inv_base_assoc 📖	mathematical	—	CategoryTheory.CategoryStruct.comp TopCat CategoryTheory.Category.toCategoryStruct TopCat.instCategory AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom' CategoryTheory.Iso.hom AlgebraicGeometry.Scheme instCategory CategoryTheory.Iso.inv	—	CategoryTheory.Category.assoc CategoryTheory.Category.id_comp Mathlib.Tactic.Reassoc.eq_whisker' hom_base_inv_base
hom_inv_apply 📖	mathematical	—	DFunLike.coe AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace TopCat.carrier CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap TopCat.str ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom' CategoryTheory.Iso.inv AlgebraicGeometry.Scheme instCategory CategoryTheory.Iso.hom	—	CategoryTheory.Iso.hom_inv_id
homeoOfIso_apply 📖	mathematical	—	DFunLike.coe Homeomorph TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier EquivLike.toFunLike Homeomorph.instEquivLike homeoOfIso CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap TopCat.str ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom' CategoryTheory.Iso.hom AlgebraicGeometry.Scheme instCategory	—	—
homeoOfIso_symm 📖	mathematical	—	Homeomorph.symm TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier homeoOfIso CategoryTheory.Iso.symm AlgebraicGeometry.Scheme instCategory	—	—
id_app 📖	mathematical	—	Hom.app CategoryTheory.CategoryStruct.id AlgebraicGeometry.Scheme CategoryTheory.Category.toCategoryStruct instCategory CommRingCat CommRingCat.instCategory CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op Opens	—	Hom.id_app
id_appTop 📖	mathematical	—	Hom.appTop CategoryTheory.CategoryStruct.id AlgebraicGeometry.Scheme CategoryTheory.Category.toCategoryStruct instCategory CommRingCat CommRingCat.instCategory CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op Opens Top.top OrderTop.toTop Preorder.toLE SemilatticeSup.toPartialOrder Lattice.toSemilatticeSup CompleteLattice.toLattice TopologicalSpace.Opens.instCompleteLattice AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier BoundedOrder.toOrderTop CompleteLattice.toBoundedOrder	—	Hom.id_appTop
instFaithfulLocallyRingedSpaceForgetToLocallyRingedSpace 📖	mathematical	—	CategoryTheory.Functor.Faithful AlgebraicGeometry.Scheme instCategory AlgebraicGeometry.LocallyRingedSpace AlgebraicGeometry.LocallyRingedSpace.instCategory forgetToLocallyRingedSpace	—	CategoryTheory.Functor.FullyFaithful.faithful
instFullLocallyRingedSpaceForgetToLocallyRingedSpace 📖	mathematical	—	CategoryTheory.Functor.Full AlgebraicGeometry.Scheme instCategory AlgebraicGeometry.LocallyRingedSpace AlgebraicGeometry.LocallyRingedSpace.instCategory forgetToLocallyRingedSpace	—	CategoryTheory.Functor.FullyFaithful.full
instSubsingletonCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensBot 📖	mathematical	—	CommRingCat.carrier CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op Opens Bot.bot OrderBot.toBot Preorder.toLE SemilatticeSup.toPartialOrder Lattice.toSemilatticeSup GeneralizedHeytingAlgebra.toLattice HeytingAlgebra.toGeneralizedHeytingAlgebra Order.Frame.toHeytingAlgebra TopologicalSpace.Opens.instFrame AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier HeytingAlgebra.toOrderBot	—	CommRingCat.subsingleton_of_isTerminal
inv_app 📖	mathematical	—	Hom.app CategoryTheory.inv AlgebraicGeometry.Scheme instCategory CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op Opens TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom' AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier CategoryTheory.Functor.map Quiver.Hom.op CategoryTheory.CategoryStruct.toQuiver CategoryTheory.eqToHom Hom.instIsIsoCommRingCatApp	—	Hom.inv_app
inv_appTop 📖	mathematical	—	Hom.appTop CategoryTheory.inv AlgebraicGeometry.Scheme instCategory CommRingCat CommRingCat.instCategory CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op Opens Top.top OrderTop.toTop Preorder.toLE SemilatticeSup.toPartialOrder Lattice.toSemilatticeSup CompleteLattice.toLattice TopologicalSpace.Opens.instCompleteLattice AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier BoundedOrder.toOrderTop CompleteLattice.toBoundedOrder Hom.instIsIsoCommRingCatApp	—	Hom.inv_appTop
inv_base_hom_base 📖	mathematical	—	CategoryTheory.CategoryStruct.comp TopCat CategoryTheory.Category.toCategoryStruct TopCat.instCategory AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom' CategoryTheory.Iso.inv AlgebraicGeometry.Scheme instCategory CategoryTheory.Iso.hom CategoryTheory.CategoryStruct.id	—	AlgebraicGeometry.LocallyRingedSpace.iso_inv_base_hom_base
inv_base_hom_base_assoc 📖	mathematical	—	CategoryTheory.CategoryStruct.comp TopCat CategoryTheory.Category.toCategoryStruct TopCat.instCategory AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom' CategoryTheory.Iso.inv AlgebraicGeometry.Scheme instCategory CategoryTheory.Iso.hom	—	CategoryTheory.Category.assoc CategoryTheory.Category.id_comp Mathlib.Tactic.Reassoc.eq_whisker' inv_base_hom_base
inv_hom_apply 📖	mathematical	—	DFunLike.coe AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace TopCat.carrier CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap TopCat.str ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom' CategoryTheory.Iso.hom AlgebraicGeometry.Scheme instCategory CategoryTheory.Iso.inv	—	CategoryTheory.Iso.inv_hom_id
isEmpty_of_commSq 📖	mathematical	CategoryTheory.CommSq AlgebraicGeometry.Scheme instCategory Disjoint Set TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace OmegaCompletePartialOrder.toPartialOrder CompleteLattice.instOmegaCompletePartialOrder CompleteBooleanAlgebra.toCompleteLattice CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra Set.instCompleteAtomicBooleanAlgebra HeytingAlgebra.toOrderBot Order.Frame.toHeytingAlgebra CompleteDistribLattice.toFrame CompleteBooleanAlgebra.toCompleteDistribLattice Set.range DFunLike.coe CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap TopCat.str ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom'	IsEmpty TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace	—	Eq.le Set.disjoint_iff_inter_eq_empty CategoryTheory.CommSq.w
iso_hom_base_inv_base 📖	mathematical	—	CategoryTheory.CategoryStruct.comp TopCat CategoryTheory.Category.toCategoryStruct TopCat.instCategory AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom' CategoryTheory.Iso.hom AlgebraicGeometry.Scheme instCategory CategoryTheory.Iso.inv CategoryTheory.CategoryStruct.id	—	hom_base_inv_base
iso_hom_base_inv_base_apply 📖	mathematical	—	DFunLike.coe AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace TopCat.carrier CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap TopCat.str ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom' CategoryTheory.Iso.inv AlgebraicGeometry.Scheme instCategory CategoryTheory.Iso.hom	—	hom_inv_apply
iso_inv_base_hom_base 📖	mathematical	—	CategoryTheory.CategoryStruct.comp TopCat CategoryTheory.Category.toCategoryStruct TopCat.instCategory AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom' CategoryTheory.Iso.inv AlgebraicGeometry.Scheme instCategory CategoryTheory.Iso.hom CategoryTheory.CategoryStruct.id	—	inv_base_hom_base
iso_inv_base_hom_base_apply 📖	mathematical	—	DFunLike.coe AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace TopCat.carrier CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap TopCat.str ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom' CategoryTheory.Iso.hom AlgebraicGeometry.Scheme instCategory CategoryTheory.Iso.inv	—	inv_hom_apply
le_iff_specializes 📖	mathematical	—	TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace Preorder.toLE instPreorderCarrierCarrierCommRingCat Specializes AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier	—	—
local_affine 📖	mathematical	—	TopologicalSpace.OpenNhds AlgebraicGeometry.LocallyRingedSpace.toTopCat toLocallyRingedSpace CommRingCat CategoryTheory.Iso AlgebraicGeometry.LocallyRingedSpace AlgebraicGeometry.LocallyRingedSpace.instCategory AlgebraicGeometry.LocallyRingedSpace.restrict CategoryTheory.Functor.obj TopologicalSpace.Opens TopCat.carrier TopCat.str Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder TopCat TopCat.instCategory TopologicalSpace.Opens.toTopCat SetLike.instMembership TopologicalSpace.Opens.instSetLike TopologicalSpace.Opens.inclusion' TopologicalSpace.OpenNhds.isOpenEmbedding Opposite CategoryTheory.Category.opposite CommRingCat.instCategory AlgebraicGeometry.Spec.toLocallyRingedSpace Opposite.op	—	—
mem_basicOpen 📖	mathematical	TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace Opens SetLike.instMembership TopologicalSpace.Opens.instSetLike AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier	TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace Opens SetLike.instMembership TopologicalSpace.Opens.instSetLike AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier basicOpen IsUnit CommRingCat.carrier TopCat.Presheaf.stalk CommRingCat.Colimits.hasColimits_commRingCat AlgebraicGeometry.PresheafedSpace.presheaf MonoidWithZero.toMonoid Semiring.toMonoidWithZero CommSemiring.toSemiring CommRing.toCommSemiring CommRingCat.commRing DFunLike.coe CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder Opposite.op CategoryTheory.ConcreteCategory.hom RingHom Semiring.toNonAssocSemiring RingHom.instFunLike CommRingCat.instConcreteCategoryRingHomCarrier TopCat.Presheaf.germ	—	AlgebraicGeometry.RingedSpace.mem_basicOpen
mem_basicOpen' 📖	mathematical	—	TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace Opens SetLike.instMembership TopologicalSpace.Opens.instSetLike AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier basicOpen IsUnit CommRingCat.carrier TopCat.Presheaf.stalk CommRingCat.Colimits.hasColimits_commRingCat AlgebraicGeometry.PresheafedSpace.presheaf MonoidWithZero.toMonoid Semiring.toMonoidWithZero CommSemiring.toSemiring CommRing.toCommSemiring CommRingCat.commRing DFunLike.coe CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder Opposite.op CategoryTheory.ConcreteCategory.hom RingHom Semiring.toNonAssocSemiring RingHom.instFunLike CommRingCat.instConcreteCategoryRingHomCarrier TopCat.Presheaf.germ	—	AlgebraicGeometry.RingedSpace.mem_basicOpen
mem_basicOpen'' 📖	mathematical	—	TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace Opens SetLike.instMembership TopologicalSpace.Opens.instSetLike AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier basicOpen IsUnit CommRingCat.carrier TopCat.Presheaf.stalk CommRingCat.Colimits.hasColimits_commRingCat AlgebraicGeometry.PresheafedSpace.presheaf MonoidWithZero.toMonoid Semiring.toMonoidWithZero CommSemiring.toSemiring CommRing.toCommSemiring CommRingCat.commRing DFunLike.coe CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder Opposite.op CategoryTheory.ConcreteCategory.hom RingHom Semiring.toNonAssocSemiring RingHom.instFunLike CommRingCat.instConcreteCategoryRingHomCarrier TopCat.Presheaf.germ	—	—
mem_basicOpen_top 📖	mathematical	—	TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace Opens SetLike.instMembership TopologicalSpace.Opens.instSetLike AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier basicOpen Top.top OrderTop.toTop Preorder.toLE PartialOrder.toPreorder SemilatticeSup.toPartialOrder Lattice.toSemilatticeSup CompleteLattice.toLattice TopologicalSpace.Opens.instCompleteLattice BoundedOrder.toOrderTop CompleteLattice.toBoundedOrder IsUnit CommRingCat.carrier TopCat.Presheaf.stalk CommRingCat.Colimits.hasColimits_commRingCat AlgebraicGeometry.PresheafedSpace.presheaf MonoidWithZero.toMonoid Semiring.toMonoidWithZero CommSemiring.toSemiring CommRing.toCommSemiring CommRingCat.commRing DFunLike.coe CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory TopologicalSpace.Opens.instPartialOrder Opposite.op CategoryTheory.ConcreteCategory.hom RingHom Semiring.toNonAssocSemiring RingHom.instFunLike CommRingCat.instConcreteCategoryRingHomCarrier TopCat.Presheaf.germ	—	AlgebraicGeometry.RingedSpace.mem_top_basicOpen
mem_zeroLocus_iff 📖	mathematical	—	TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace Set Set.instMembership zeroLocus Opens SetLike.instMembership TopologicalSpace.Opens.instSetLike AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier basicOpen	—	AlgebraicGeometry.RingedSpace.mem_zeroLocus_iff
preimage_basicOpen 📖	mathematical	—	CategoryTheory.Functor.obj Opens Preorder.smallCategory TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace TopCat.str PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom' basicOpen AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier DFunLike.coe Opposite CategoryTheory.Category.opposite AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op CommRingCat.carrier CategoryTheory.ConcreteCategory.hom RingHom Semiring.toNonAssocSemiring CommSemiring.toSemiring CommRing.toCommSemiring CommRingCat.commRing RingHom.instFunLike CommRingCat.instConcreteCategoryRingHomCarrier Hom.app	—	AlgebraicGeometry.LocallyRingedSpace.preimage_basicOpen
preimage_basicOpen_top 📖	mathematical	—	CategoryTheory.Functor.obj Opens Preorder.smallCategory TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace TopCat.str PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom' basicOpen Top.top OrderTop.toTop Preorder.toLE SemilatticeSup.toPartialOrder Lattice.toSemilatticeSup CompleteLattice.toLattice TopologicalSpace.Opens.instCompleteLattice AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier BoundedOrder.toOrderTop CompleteLattice.toBoundedOrder DFunLike.coe Opposite CategoryTheory.Category.opposite AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op CommRingCat.carrier CategoryTheory.ConcreteCategory.hom RingHom Semiring.toNonAssocSemiring CommSemiring.toSemiring CommRing.toCommSemiring CommRingCat.commRing RingHom.instFunLike CommRingCat.instConcreteCategoryRingHomCarrier Hom.appTop	—	preimage_basicOpen
preimage_comp 📖	mathematical	—	CategoryTheory.Functor.obj Opens Preorder.smallCategory TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace TopCat.str PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom' CategoryTheory.CategoryStruct.comp AlgebraicGeometry.Scheme CategoryTheory.Category.toCategoryStruct instCategory AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier	—	Hom.comp_preimage
preimage_zeroLocus 📖	mathematical	—	Set.preimage TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace DFunLike.coe CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap TopCat.str ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom' zeroLocus CategoryTheory.Functor.obj Opens Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier TopologicalSpace.Opens.map Set.image CommRingCat.carrier Opposite TopologicalSpace.Opens CategoryTheory.Category.opposite AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op RingHom Semiring.toNonAssocSemiring CommSemiring.toSemiring CommRing.toCommSemiring CommRingCat.commRing RingHom.instFunLike CommRingCat.Hom.hom Hom.app	—	Set.ext
presheaf_map_eqToHom_op 📖	mathematical	—	CategoryTheory.Functor.map CategoryTheory.eqToHom CategoryTheory.Category.toCategoryStruct CategoryTheory.Functor.obj	—	CategoryTheory.eqToHom_map
stalkMap_comp 📖	mathematical	—	Hom.stalkMap CategoryTheory.CategoryStruct.comp AlgebraicGeometry.Scheme CategoryTheory.Category.toCategoryStruct instCategory CommRingCat CommRingCat.instCategory TopCat.Presheaf.stalk CommRingCat.Colimits.hasColimits_commRingCat AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.PresheafedSpace.presheaf DFunLike.coe TopCat.carrier CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap TopCat.str ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom'	—	Hom.stalkMap_comp
stalkMap_congr 📖	mathematical	—	CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory TopCat.Presheaf.stalk CommRingCat.Colimits.hasColimits_commRingCat AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.PresheafedSpace.presheaf DFunLike.coe TopCat.carrier CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap TopCat.str ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom' Hom.stalkMap CategoryTheory.Iso.hom TopCat.Presheaf.stalkCongr Inseparable.of_eq Quiver.Hom AlgebraicGeometry.Scheme CategoryTheory.CategoryStruct.toQuiver instCategory	—	Hom.stalkMap_congr
stalkMap_congr_assoc 📖	mathematical	—	CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory TopCat.Presheaf.stalk CommRingCat.Colimits.hasColimits_commRingCat AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.PresheafedSpace.presheaf DFunLike.coe ContinuousMap TopCat.carrier TopCat.str CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom' Hom.stalkMap CategoryTheory.Iso.hom TopCat.Presheaf.stalkCongr Inseparable.of_eq Quiver.Hom AlgebraicGeometry.Scheme CategoryTheory.CategoryStruct.toQuiver instCategory	—	Hom.stalkMap_congr_assoc
stalkMap_congr_hom 📖	mathematical	—	Hom.stalkMap CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory TopCat.Presheaf.stalk CommRingCat.Colimits.hasColimits_commRingCat AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.PresheafedSpace.presheaf DFunLike.coe TopCat.carrier CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap TopCat.str ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom' CategoryTheory.Iso.hom TopCat.Presheaf.stalkCongr Inseparable.of_eq Quiver.Hom AlgebraicGeometry.Scheme CategoryTheory.CategoryStruct.toQuiver instCategory	—	Hom.stalkMap_congr_hom
stalkMap_congr_hom_assoc 📖	mathematical	—	CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory TopCat.Presheaf.stalk CommRingCat.Colimits.hasColimits_commRingCat AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.PresheafedSpace.presheaf DFunLike.coe ContinuousMap TopCat.carrier TopCat.str CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom' Hom.stalkMap CategoryTheory.Iso.hom TopCat.Presheaf.stalkCongr Inseparable.of_eq Quiver.Hom AlgebraicGeometry.Scheme CategoryTheory.CategoryStruct.toQuiver instCategory	—	Hom.stalkMap_congr_hom_assoc
stalkMap_congr_point 📖	mathematical	—	CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory TopCat.Presheaf.stalk CommRingCat.Colimits.hasColimits_commRingCat AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.PresheafedSpace.presheaf DFunLike.coe TopCat.carrier CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap TopCat.str ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom' Hom.stalkMap CategoryTheory.Iso.hom TopCat.Presheaf.stalkCongr Inseparable.of_eq	—	Hom.stalkMap_congr_point
stalkMap_congr_point_assoc 📖	mathematical	—	CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory TopCat.Presheaf.stalk CommRingCat.Colimits.hasColimits_commRingCat AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.PresheafedSpace.presheaf DFunLike.coe ContinuousMap TopCat.carrier TopCat.str CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom' Hom.stalkMap CategoryTheory.Iso.hom TopCat.Presheaf.stalkCongr Inseparable.of_eq	—	Hom.stalkMap_congr_point_assoc
stalkMap_germ 📖	mathematical	TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace Opens SetLike.instMembership TopologicalSpace.Opens.instSetLike AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier DFunLike.coe CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap TopCat.str ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom'	CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op TopCat.Presheaf.stalk CommRingCat.Colimits.hasColimits_commRingCat DFunLike.coe CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom' TopCat.Presheaf.germ Hom.stalkMap Opens AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier TopologicalSpace.Opens.map Hom.app	—	Hom.germ_stalkMap
stalkMap_germ_apply 📖	mathematical	TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace Opens SetLike.instMembership TopologicalSpace.Opens.instSetLike AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier DFunLike.coe CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap TopCat.str ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom'	DFunLike.coe TopCat.Presheaf.stalk CommRingCat CommRingCat.instCategory CommRingCat.Colimits.hasColimits_commRingCat AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.PresheafedSpace.presheaf TopCat.carrier CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap TopCat.str ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom' CommRingCat.carrier RingHom Semiring.toNonAssocSemiring CommSemiring.toSemiring CommRing.toCommSemiring CommRingCat.commRing RingHom.instFunLike CommRingCat.instConcreteCategoryRingHomCarrier Hom.stalkMap CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder Opposite.op TopCat.Presheaf.germ Opens AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier TopologicalSpace.Opens.map Hom.app	—	Hom.germ_stalkMap_apply
stalkMap_germ_assoc 📖	mathematical	TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace Opens SetLike.instMembership TopologicalSpace.Opens.instSetLike AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier DFunLike.coe CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap TopCat.str ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom'	CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op TopCat.Presheaf.stalk CommRingCat.Colimits.hasColimits_commRingCat DFunLike.coe ContinuousMap CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom' TopCat.Presheaf.germ Hom.stalkMap Opens AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier TopologicalSpace.Opens.map Hom.app	—	Hom.germ_stalkMap_assoc
stalkMap_hom_inv 📖	mathematical	—	CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory TopCat.Presheaf.stalk CommRingCat.Colimits.hasColimits_commRingCat AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.PresheafedSpace.presheaf DFunLike.coe TopCat.carrier CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap TopCat.str ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom' CategoryTheory.Iso.hom AlgebraicGeometry.Scheme instCategory CategoryTheory.Iso.inv Hom.stalkMap TopCat.Presheaf.stalkCongr Inseparable.of_eq	—	Hom.stalkMap_hom_inv
stalkMap_hom_inv_apply 📖	mathematical	—	DFunLike.coe TopCat.Presheaf.stalk CommRingCat CommRingCat.instCategory CommRingCat.Colimits.hasColimits_commRingCat AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.PresheafedSpace.presheaf TopCat.carrier CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap TopCat.str ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom' CategoryTheory.Iso.inv AlgebraicGeometry.Scheme instCategory CommRingCat.carrier RingHom Semiring.toNonAssocSemiring CommSemiring.toSemiring CommRing.toCommSemiring CommRingCat.commRing RingHom.instFunLike CommRingCat.instConcreteCategoryRingHomCarrier Hom.stalkMap CategoryTheory.Iso.hom TopCat.Presheaf.stalkCongr Inseparable.of_eq	—	Hom.stalkMap_hom_inv_apply
stalkMap_hom_inv_assoc 📖	mathematical	—	CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory TopCat.Presheaf.stalk CommRingCat.Colimits.hasColimits_commRingCat AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.PresheafedSpace.presheaf DFunLike.coe ContinuousMap TopCat.carrier TopCat.str CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom' CategoryTheory.Iso.hom AlgebraicGeometry.Scheme instCategory CategoryTheory.Iso.inv Hom.stalkMap TopCat.Presheaf.stalkCongr Inseparable.of_eq	—	Hom.stalkMap_hom_inv_assoc
stalkMap_id 📖	mathematical	—	Hom.stalkMap CategoryTheory.CategoryStruct.id AlgebraicGeometry.Scheme CategoryTheory.Category.toCategoryStruct instCategory CommRingCat CommRingCat.instCategory TopCat.Presheaf.stalk CommRingCat.Colimits.hasColimits_commRingCat AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.PresheafedSpace.presheaf	—	Hom.stalkMap_id
stalkMap_inv_hom 📖	mathematical	—	CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory TopCat.Presheaf.stalk CommRingCat.Colimits.hasColimits_commRingCat AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.PresheafedSpace.presheaf DFunLike.coe TopCat.carrier CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap TopCat.str ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom' CategoryTheory.Iso.inv AlgebraicGeometry.Scheme instCategory CategoryTheory.Iso.hom Hom.stalkMap TopCat.Presheaf.stalkCongr Inseparable.of_eq	—	Hom.stalkMap_inv_hom
stalkMap_inv_hom_apply 📖	mathematical	—	DFunLike.coe TopCat.Presheaf.stalk CommRingCat CommRingCat.instCategory CommRingCat.Colimits.hasColimits_commRingCat AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.PresheafedSpace.presheaf TopCat.carrier CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap TopCat.str ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom' CategoryTheory.Iso.hom AlgebraicGeometry.Scheme instCategory CommRingCat.carrier RingHom Semiring.toNonAssocSemiring CommSemiring.toSemiring CommRing.toCommSemiring CommRingCat.commRing RingHom.instFunLike CommRingCat.instConcreteCategoryRingHomCarrier Hom.stalkMap CategoryTheory.Iso.inv TopCat.Presheaf.stalkCongr Inseparable.of_eq	—	Hom.stalkMap_inv_hom_apply
stalkMap_inv_hom_assoc 📖	mathematical	—	CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory TopCat.Presheaf.stalk CommRingCat.Colimits.hasColimits_commRingCat AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.PresheafedSpace.presheaf DFunLike.coe ContinuousMap TopCat.carrier TopCat.str CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom' CategoryTheory.Iso.inv AlgebraicGeometry.Scheme instCategory CategoryTheory.Iso.hom Hom.stalkMap TopCat.Presheaf.stalkCongr Inseparable.of_eq	—	Hom.stalkMap_inv_hom_assoc
stalkSpecializes_stalkMap 📖	mathematical	Specializes TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier	CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory TopCat.Presheaf.stalk CommRingCat.Colimits.hasColimits_commRingCat AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.PresheafedSpace.presheaf DFunLike.coe ContinuousMap TopCat.carrier TopCat.str ContinuousMap.instFunLike TopCat.Hom.hom AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom' TopCat.Presheaf.stalkSpecializes ContinuousMap.map_specializes Hom.stalkMap CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory TopCat.instConcreteCategoryContinuousMapCarrier	—	Hom.stalkSpecializes_stalkMap
stalkSpecializes_stalkMap_apply 📖	mathematical	Specializes TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier	DFunLike.coe TopCat.Presheaf.stalk CommRingCat CommRingCat.instCategory CommRingCat.Colimits.hasColimits_commRingCat AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.PresheafedSpace.presheaf TopCat.carrier CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap TopCat.str ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom' CommRingCat.carrier RingHom Semiring.toNonAssocSemiring CommSemiring.toSemiring CommRing.toCommSemiring CommRingCat.commRing RingHom.instFunLike CommRingCat.instConcreteCategoryRingHomCarrier Hom.stalkMap TopCat.Hom.hom TopCat.Presheaf.stalkSpecializes ContinuousMap.map_specializes	—	Hom.stalkSpecializes_stalkMap_apply
stalkSpecializes_stalkMap_assoc 📖	mathematical	Specializes TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier	CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory TopCat.Presheaf.stalk CommRingCat.Colimits.hasColimits_commRingCat AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.PresheafedSpace.presheaf DFunLike.coe ContinuousMap TopCat.carrier TopCat.str ContinuousMap.instFunLike TopCat.Hom.hom AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom Hom.toLRSHom' TopCat.Presheaf.stalkSpecializes ContinuousMap.map_specializes Hom.stalkMap	—	Hom.stalkSpecializes_stalkMap_assoc
toOpen_eq 📖	mathematical	—	CommRingCat.ofHom CommRingCat.carrier CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PrimeSpectrum.Top CommRingCat.commRing TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder CommRingCat CommRingCat.instCategory TopCat.Sheaf.presheaf AlgebraicGeometry.Spec.structureSheaf Opposite.op algebraMap CommRing.toCommSemiring CommSemiring.toSemiring AlgebraicGeometry.StructureSheaf.openAlgebra CategoryTheory.CategoryStruct.comp CategoryTheory.Category.toCategoryStruct AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.Spec AlgebraicGeometry.PresheafedSpace.presheaf Opens Top.top OrderTop.toTop Preorder.toLE SemilatticeSup.toPartialOrder Lattice.toSemilatticeSup CompleteLattice.toLattice TopologicalSpace.Opens.instCompleteLattice AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier BoundedOrder.toOrderTop CompleteLattice.toBoundedOrder CategoryTheory.Iso.inv ΓSpecIso CategoryTheory.Functor.map Quiver.Hom.op CategoryTheory.CategoryStruct.toQuiver CategoryTheory.homOfLE le_top	—	—
zeroLocus_def 📖	mathematical	—	zeroLocus Set.iInter TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace CommRingCat.carrier CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op Opens Set Set.instMembership Compl.compl Set.instCompl TopologicalSpace.Opens.carrier AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier basicOpen	—	—
zeroLocus_empty_eq_univ 📖	mathematical	—	zeroLocus Set CommRingCat.carrier CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op Opens Set.instEmptyCollection Set.univ	—	AlgebraicGeometry.RingedSpace.zeroLocus_empty_eq_univ
zeroLocus_iUnion 📖	mathematical	—	zeroLocus Set.iUnion CommRingCat.carrier CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op Opens Set.iInter	—	Set.iInter_congr_Prop Set.iInter_exists Set.iInter_comm
zeroLocus_isClosed 📖	mathematical	—	IsClosed TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier zeroLocus	—	AlgebraicGeometry.RingedSpace.zeroLocus_isClosed
zeroLocus_map 📖	mathematical	Opens Preorder.toLE PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier	zeroLocus Set.image CommRingCat.carrier CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op Opens DFunLike.coe RingHom Semiring.toNonAssocSemiring CommSemiring.toSemiring CommRing.toCommSemiring CommRingCat.commRing RingHom.instFunLike CommRingCat.Hom.hom CategoryTheory.Functor.map Quiver.Hom.op CategoryTheory.CategoryStruct.toQuiver CategoryTheory.Category.toCategoryStruct AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier CategoryTheory.homOfLE Set Set.instUnion Compl.compl Set.instCompl SetLike.coe TopologicalSpace.Opens.instSetLike	—	Set.ext basicOpen_res
zeroLocus_map_of_eq 📖	mathematical	—	zeroLocus Set.image CommRingCat.carrier CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op Opens DFunLike.coe RingHom Semiring.toNonAssocSemiring CommSemiring.toSemiring CommRing.toCommSemiring CommRingCat.commRing RingHom.instFunLike CommRingCat.Hom.hom CategoryTheory.Functor.map Quiver.Hom.op CategoryTheory.CategoryStruct.toQuiver CategoryTheory.Category.toCategoryStruct AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier CategoryTheory.eqToHom	—	Set.ext Set.image_congr CategoryTheory.eqToHom_op basicOpen_res_eq CategoryTheory.instIsIsoEqToHom
zeroLocus_mono 📖	mathematical	Set CommRingCat.carrier CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op Opens Set.instHasSubset	Set TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace Set.instHasSubset zeroLocus	—	—
zeroLocus_mul 📖	mathematical	—	zeroLocus SetLike.coe Ideal CommRingCat.carrier CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op Opens CommSemiring.toSemiring CommRing.toCommSemiring CommRingCat.commRing Submodule.setLike NonUnitalNonAssocSemiring.toAddCommMonoid NonAssocSemiring.toNonUnitalNonAssocSemiring Semiring.toNonAssocSemiring Semiring.toModule Submodule.mul IsScalarTower.right Algebra.id Set Set.instUnion	—	IsScalarTower.right zeroLocus_setMul zeroLocus_span IsScalarTower.left Ideal.span_mul_span Ideal.instIsTwoSided_1 Ideal.span_eq
zeroLocus_radical 📖	mathematical	—	zeroLocus SetLike.coe Ideal CommRingCat.carrier CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op Opens CommSemiring.toSemiring CommRing.toCommSemiring CommRingCat.commRing Submodule.setLike NonUnitalNonAssocSemiring.toAddCommMonoid NonAssocSemiring.toNonUnitalNonAssocSemiring Semiring.toNonAssocSemiring Semiring.toModule Ideal.radical	—	HasSubset.Subset.antisymm Set.instAntisymmSubset zeroLocus_mono Ideal.le_radical pow_zero mul_one Ideal.mul_mem_left basicOpen_pow
zeroLocus_setMul 📖	mathematical	—	zeroLocus Set CommRingCat.carrier CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op Opens Set.mul Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing CommRing.toNonUnitalCommRing CommRingCat.commRing Set.instUnion	—	Set.biInter_image2 Set.iInter_congr_Prop basicOpen_mul Set.compl_inter
zeroLocus_singleton 📖	mathematical	—	zeroLocus CommRingCat.carrier CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op Opens Set Set.instSingletonSet Compl.compl Set.instCompl SetLike.coe TopologicalSpace.Opens.instSetLike AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier basicOpen	—	AlgebraicGeometry.RingedSpace.zeroLocus_singleton
zeroLocus_span 📖	mathematical	—	zeroLocus SetLike.coe Ideal CommRingCat.carrier CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op Opens CommSemiring.toSemiring CommRing.toCommSemiring CommRingCat.commRing Submodule.setLike NonUnitalNonAssocSemiring.toAddCommMonoid NonAssocSemiring.toNonUnitalNonAssocSemiring Semiring.toNonAssocSemiring Semiring.toModule Ideal.span	—	Set.ext Ideal.subset_span Submodule.span_induction basicOpen_zero basicOpen_add_le basicOpen_mul
zeroLocus_univ 📖	mathematical	—	zeroLocus Set.univ CommRingCat.carrier CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op Opens Compl.compl Set Set.instCompl SetLike.coe TopologicalSpace.Opens.instSetLike AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier	—	Set.ext basicOpen_le basicOpen_of_isUnit isUnit_one
ΓSpecIso_inv 📖	mathematical	—	CategoryTheory.Iso.inv CommRingCat CommRingCat.instCategory CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.Spec TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op Opens Top.top OrderTop.toTop Preorder.toLE SemilatticeSup.toPartialOrder Lattice.toSemilatticeSup CompleteLattice.toLattice TopologicalSpace.Opens.instCompleteLattice AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier BoundedOrder.toOrderTop CompleteLattice.toBoundedOrder ΓSpecIso CommRingCat.ofHom CommRingCat.carrier AlgebraicGeometry.PrimeSpectrum.Top CommRingCat.commRing CategoryTheory.types CategoryTheory.ObjectProperty.FullSubcategory.obj CategoryTheory.Functor CategoryTheory.Functor.category CategoryTheory.Presheaf.IsSheaf Opens.grothendieckTopology AlgebraicGeometry.structureSheafInType Ring.toAddCommGroup CommRing.toRing Semiring.toModule CommSemiring.toSemiring CommRing.toCommSemiring AlgebraicGeometry.instCommRingObjOppositeOpensCarrierTopObjFunctorTypeIsSheafGrothendieckTopologyStructureSheafInType Algebra.id algebraMap AlgebraicGeometry.instAlgebraObjOppositeOpensCarrierTopObjFunctorTypeIsSheafGrothendieckTopologyStructureSheafInType	—	—
ΓSpecIso_inv_naturality 📖	mathematical	—	CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.Spec TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op Opens Top.top OrderTop.toTop Preorder.toLE SemilatticeSup.toPartialOrder Lattice.toSemilatticeSup CompleteLattice.toLattice TopologicalSpace.Opens.instCompleteLattice AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier BoundedOrder.toOrderTop CompleteLattice.toBoundedOrder CategoryTheory.Iso.inv ΓSpecIso Hom.appTop AlgebraicGeometry.Spec.map	—	CategoryTheory.NatTrans.naturality
ΓSpecIso_inv_naturality_assoc 📖	mathematical	—	CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.Spec TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op Opens Top.top OrderTop.toTop Preorder.toLE SemilatticeSup.toPartialOrder Lattice.toSemilatticeSup CompleteLattice.toLattice TopologicalSpace.Opens.instCompleteLattice AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier BoundedOrder.toOrderTop CompleteLattice.toBoundedOrder CategoryTheory.Iso.inv ΓSpecIso Hom.appTop AlgebraicGeometry.Spec.map	—	CategoryTheory.Category.assoc Mathlib.Tactic.Reassoc.eq_whisker' ΓSpecIso_inv_naturality
ΓSpecIso_naturality 📖	mathematical	—	CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.Spec TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op Opens Top.top OrderTop.toTop Preorder.toLE SemilatticeSup.toPartialOrder Lattice.toSemilatticeSup CompleteLattice.toLattice TopologicalSpace.Opens.instCompleteLattice AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier BoundedOrder.toOrderTop CompleteLattice.toBoundedOrder Hom.appTop AlgebraicGeometry.Spec.map CategoryTheory.Iso.hom ΓSpecIso	—	CategoryTheory.NatTrans.naturality
ΓSpecIso_naturality_assoc 📖	mathematical	—	CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace AlgebraicGeometry.Spec TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op Opens Top.top OrderTop.toTop Preorder.toLE SemilatticeSup.toPartialOrder Lattice.toSemilatticeSup CompleteLattice.toLattice TopologicalSpace.Opens.instCompleteLattice AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier BoundedOrder.toOrderTop CompleteLattice.toBoundedOrder Hom.appTop AlgebraicGeometry.Spec.map CategoryTheory.Iso.hom ΓSpecIso	—	CategoryTheory.Category.assoc Mathlib.Tactic.Reassoc.eq_whisker' ΓSpecIso_naturality
Γ_def 📖	mathematical	—	Γ CategoryTheory.Functor.comp Opposite AlgebraicGeometry.Scheme CategoryTheory.Category.opposite instCategory AlgebraicGeometry.LocallyRingedSpace AlgebraicGeometry.LocallyRingedSpace.instCategory CommRingCat CommRingCat.instCategory CategoryTheory.Functor.op forgetToLocallyRingedSpace AlgebraicGeometry.LocallyRingedSpace.Γ	—	—
Γ_map 📖	mathematical	—	CategoryTheory.Functor.map Opposite AlgebraicGeometry.Scheme CategoryTheory.Category.opposite instCategory CommRingCat CommRingCat.instCategory Γ Hom.appTop Opposite.unop Quiver.Hom.unop CategoryTheory.CategoryStruct.toQuiver CategoryTheory.Category.toCategoryStruct	—	—
Γ_map_op 📖	mathematical	—	CategoryTheory.Functor.map Opposite AlgebraicGeometry.Scheme CategoryTheory.Category.opposite instCategory CommRingCat CommRingCat.instCategory Γ Opposite.op Quiver.Hom.op CategoryTheory.CategoryStruct.toQuiver CategoryTheory.Category.toCategoryStruct Hom.appTop	—	—
Γ_obj 📖	mathematical	—	CategoryTheory.Functor.obj Opposite AlgebraicGeometry.Scheme CategoryTheory.Category.opposite instCategory CommRingCat CommRingCat.instCategory Γ TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace Opposite.unop TopCat.str Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op Opens Top.top OrderTop.toTop Preorder.toLE SemilatticeSup.toPartialOrder Lattice.toSemilatticeSup CompleteLattice.toLattice TopologicalSpace.Opens.instCompleteLattice AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier BoundedOrder.toOrderTop CompleteLattice.toBoundedOrder	—	—
Γ_obj_op 📖	mathematical	—	CategoryTheory.Functor.obj Opposite AlgebraicGeometry.Scheme CategoryTheory.Category.opposite instCategory CommRingCat CommRingCat.instCategory Γ Opposite.op TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace toLocallyRingedSpace TopCat.str Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opens Top.top OrderTop.toTop Preorder.toLE SemilatticeSup.toPartialOrder Lattice.toSemilatticeSup CompleteLattice.toLattice TopologicalSpace.Opens.instCompleteLattice AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier BoundedOrder.toOrderTop CompleteLattice.toBoundedOrder	—	—

AlgebraicGeometry.Scheme.Hom

Definitions

Name	Category	Theorems
app 📖	CompOp	119 math math: AlgebraicGeometry.Γ_map_morphismRestrict, AlgebraicGeometry.Scheme.coprodPresheafObjIso_hom_fst_assoc, AlgebraicGeometry.Scheme.germ_stalkClosedPointTo, germ_stalkMap_assoc, app_invApp'_assoc, AlgebraicGeometry.Scheme.app_eq, AlgebraicGeometry.Scheme.Opens.toSpecΓ_naturality, AlgebraicGeometry.Scheme.inv_app, AlgebraicGeometry.isIso_ΓSpec_adjunction_unit_app_basicOpen, AlgebraicGeometry.Scheme.congr_app, AlgebraicGeometry.Scheme.coprodPresheafObjIso_hom_snd_assoc, toNormalization_app_preimage, toImage_app_injective, AlgebraicGeometry.Scheme.IdealSheafData.ideal_le_ker_glueDataObjι, AlgebraicGeometry.Scheme.eqToHom_c_app, resLE_app_top, AlgebraicGeometry.Proj.basicOpenToSpec_app_top, AlgebraicGeometry.SurjectiveOnStalks.iff_of_isAffine, AlgebraicGeometry.IsIntegralHom.isIntegral_app, AlgebraicGeometry.Scheme.IdealSheafData.ideal_map, comp_app, ι_toNormalization, AlgebraicGeometry.Scheme.comp_appLE, congr_app, AlgebraicGeometry.Scheme.Opens.toSpecΓ_naturality_assoc, id_app, germ_stalkMap_apply, app_appIso_inv_assoc, AlgebraicGeometry.IsAffineOpen.basicOpen_fromSpec_app, AlgebraicGeometry.isIso_morphismRestrict_iff_isIso_app, app_surjective, AlgebraicGeometry.Scheme.Spec_map_presheaf_map_eqToHom, AlgebraicGeometry.liftCoborder_app, AlgebraicGeometry.IsOpenImmersion.map_ΓIso_inv_assoc, appIso_inv_app_presheafMap, app_eq, comp_appLE, ker_apply, AlgebraicGeometry.Scheme.stalkMap_germ_assoc, appIso_inv_app, comp_app_assoc, AlgebraicGeometry.Scheme.comp_app, finite_app, AlgebraicGeometry.Scheme.evaluation_naturality, ideal_ker_le, AlgebraicGeometry.IsAffineOpen.fromSpec_app_self, AlgebraicGeometry.isClosedImmersion_iff_isAffineHom, preimage_basicOpen, AlgebraicGeometry.Scheme.IdealSheafData.subschemeι_app, app_invApp', comp_appLE_assoc, AlgebraicGeometry.morphismRestrict_app', AlgebraicGeometry.Scheme.evaluation_naturality_apply, AlgebraicGeometry.targetAffineLocally_affineAnd_iff, appIso_inv_app_assoc, AlgebraicGeometry.Scheme.coprodPresheafObjIso_hom_snd, AlgebraicGeometry.Scheme.homOfLE_app, naturality_assoc, AlgebraicGeometry.IsAffineOpen.fromSpec_app_self_apply, isIso_app, AlgebraicGeometry.morphismRestrict_appTop, AlgebraicGeometry.HasAffineProperty.affineAnd_iff, AlgebraicGeometry.IsOpenImmersion.app_eq_appIso_inv_app_of_comp_eq, AlgebraicGeometry.Scheme.IdealSheafData.ker_subschemeι_app, ι_toNormalization_assoc, fromNormalization_app_assoc, AlgebraicGeometry.Scheme.stalkMap_germ_apply, instIsIsoCommRingCatAppObjOpensOpensFunctor, germ_stalkMap, AlgebraicGeometry.Scheme.coprodPresheafObjIso_hom_fst, AlgebraicGeometry.IsOpenImmersion.app_ΓIso_hom_assoc, isIntegral_app, app_eq_appLE, AlgebraicGeometry.IsOpenImmersion.map_ΓIso_inv, AlgebraicGeometry.isIntegralHom_iff, AlgebraicGeometry.exists_app_map_eq_map_of_isLimit, AlgebraicGeometry.IsOpenImmersion.app_ΓIso_hom, AlgebraicGeometry.IsFinite.finite_app, AlgebraicGeometry.Scheme.germ_stalkClosedPointTo_assoc, AlgebraicGeometry.IsIntegralHom.integral_app, appIso_hom, AlgebraicGeometry.IsOpenImmersion.app_ΓIso_hom_apply, AlgebraicGeometry.Scheme.evaluation_naturality_assoc, AlgebraicGeometry.targetAffineLocally_affineAnd_iff', AlgebraicGeometry.Scheme.Opens.ι_app_self, naturality, AlgebraicGeometry.Scheme.fromSpecStalk_app, AlgebraicGeometry.Scheme.IdealSheafData.subschemeι_app_surjective, AlgebraicGeometry.Scheme.preimage_zeroLocus, AlgebraicGeometry.IsIntegralHom.hasAffineProperty, AlgebraicGeometry.Scheme.SpecMap_presheaf_map_eqToHom, AlgebraicGeometry.isFinite_iff, AlgebraicGeometry.exists_app_map_eq_zero_of_isLimit, AlgebraicGeometry.Scheme.Opens.ι_app, instIsIsoCommRingCatApp, AlgebraicGeometry.IsOpenImmersion.map_ΓIso_hom, appIso_inv_app_apply, AlgebraicGeometry.IsAffineOpen.app_basicOpen_eq_away_map, AlgebraicGeometry.Scheme.preimage_basicOpen, AlgebraicGeometry.IsAffineOpen.ΓSpecIso_hom_fromSpec_app, AlgebraicGeometry.Scheme.comp_app_assoc, AlgebraicGeometry.morphismRestrict_app, toImage_app, AlgebraicGeometry.Scheme.ofRestrict_app, AlgebraicGeometry.IsAffineOpen.fromSpec_app_of_le, app_appIso_inv, normalizationObjIso_hom_val, appLE_eq_app, AlgebraicGeometry.Scheme.IdealSheafData.ideal_map_of_isAffineHom, inv_app, app_injective, AlgebraicGeometry.Scheme.stalkMap_germ, AlgebraicGeometry.Scheme.comp_appLE_assoc, fromNormalization_app, eqToHom_app, AlgebraicGeometry.Scheme.id_app, AlgebraicGeometry.IsOpenImmersion.lift_app, AlgebraicGeometry.IsOpenImmersion.map_ΓIso_inv_apply, AlgebraicGeometry.Spec.map_app
appLE 📖	CompOp	99 math math: AlgebraicGeometry.Etale.etale_appLE, AlgebraicGeometry.LocallyQuasiFinite.quasiFinite_appLE, AlgebraicGeometry.LocallyOfFiniteType.finiteType_of_affine_subset, AlgebraicGeometry.mono_pushoutSection_of_iSup_eq, AlgebraicGeometry.SmoothOfRelativeDimension.exists_isStandardSmoothOfRelativeDimension, AlgebraicGeometry.locallyQuasiFinite_iff, AlgebraicGeometry.affineLocally_iff_forall_isAffineOpen, formallyUnramified_appLE, resLE_app_top, appLE_map', smooth_appLE, AlgebraicGeometry.Scheme.comp_appLE, AlgebraicGeometry.Scheme.ofRestrict_appLE, QuasiFiniteAt.quasiFiniteAt, AlgebraicGeometry.affineLocally_iff_affineOpens_le, AlgebraicGeometry.HasRingHomProperty.iff_of_iSup_eq_top, AlgebraicGeometry.locallyOfFiniteType_iff, AlgebraicGeometry.LocallyOfFiniteType.finiteType_appLE, AlgebraicGeometry.Spec.map_appLE, finitePresentation_appLE, AlgebraicGeometry.morphismRestrict_appLE, comp_appLE, AlgebraicGeometry.FormallyUnramified.formallyUnramified_of_affine_subset, AlgebraicGeometry.formallySmooth_stalkMap_iff, AlgebraicGeometry.HasRingHomProperty.iff_appLE, AlgebraicGeometry.isIso_pushoutSection_of_isCompact_of_flat_right_of_ringHomFlat, AlgebraicGeometry.HasRingHomProperty.appLE, map_appLE_assoc, map_appLE'_assoc, AlgebraicGeometry.smooth_iff, appIso_hom', appLE_appIso_inv, comp_appLE_assoc, AlgebraicGeometry.Proj.awayToSection_comp_appLE, AlgebraicGeometry.morphismRestrict_app', appLE_map_assoc, appIso_inv_appLE_assoc, AlgebraicGeometry.Scheme.Opens.toSpecΓ_SpecMap_appLE_assoc, AlgebraicGeometry.exists_affineOpens_le_appLE_of_appLE, AlgebraicGeometry.mono_pushoutSection_of_isCompact_of_flat_left_of_ringHomFlat, AlgebraicGeometry.exists_smooth_of_formallySmooth_stalk, AlgebraicGeometry.isIso_pushoutSection_of_iSup_eq, etale_appLE, appLE_congr, AlgebraicGeometry.IsAffineOpen.comap_primeIdealOf_appLE, appLE_appIso_inv_assoc, AlgebraicGeometry.Scheme.homOfLE_appLE, AlgebraicGeometry.Scheme.basicOpen_appLE, AlgebraicGeometry.Flat.flat_of_affine_subset, AlgebraicGeometry.IsOpenImmersion.ΓIso_inv, app_eq_appLE, map_appLE', AlgebraicGeometry.Flat.flat_appLE, AlgebraicGeometry.LocallyOfFinitePresentation.finitePresentation_appLE, finiteType_appLE, AlgebraicGeometry.mono_pushoutSection_of_isCompact_of_flat_right, AlgebraicGeometry.mono_pushoutSection_of_isCompact_of_flat_right_of_ringHomFlat, AlgebraicGeometry.HasRingHomProperty.iff_exists_appLE_locally, AlgebraicGeometry.smoothOfRelativeDimension_iff, AlgebraicGeometry.HasRingHomProperty.iff_exists_appLE, appLE_appIso_inv_apply, appLE_map'_assoc, AlgebraicGeometry.formallyUnramified_iff, AlgebraicGeometry.Smooth.exists_isStandardSmooth, AlgebraicGeometry.isIso_pushoutSection_of_isQuasiSeparated_of_flat_left, AlgebraicGeometry.isIso_pushoutSection_iff, AlgebraicGeometry.Smooth.smooth_appLE, AlgebraicGeometry.exists_basicOpen_le_appLE_of_appLE_of_isAffine, AlgebraicGeometry.IsAffineOpen.SpecMap_appLE_fromSpec_assoc, AlgebraicGeometry.IsAffineOpen.Spec_map_appLE_fromSpec, AlgebraicGeometry.LocallyOfFinitePresentation.finitePresentation_of_affine_subset, appLE_comp_appLE_assoc, AlgebraicGeometry.etale_iff, AlgebraicGeometry.IsAffineOpen.Spec_map_appLE_fromSpec_assoc, AlgebraicGeometry.mono_pushoutSection_of_isCompact_of_flat_left, AlgebraicGeometry.Scheme.Opens.ι_appLE, AlgebraicGeometry.Scheme.appLE_comp_appLE, resLE_appLE, AlgebraicGeometry.locallyOfFinitePresentation_iff, map_appLE, AlgebraicGeometry.FormallyUnramified.formallyUnramified_appLE, AlgebraicGeometry.Scheme.Opens.instIsIsoCommRingCatAppLEιTopToScheme, flat_appLE, AlgebraicGeometry.IsAffineOpen.SpecMap_appLE_fromSpec, appLE_comp_appLE, AlgebraicGeometry.isIso_pushoutSection_of_isQuasiSeparated_of_flat_right, appLE_map, normalizationObjIso_hom_val, appLE_eq_app, AlgebraicGeometry.Scheme.Opens.toSpecΓ_SpecMap_appLE, AlgebraicGeometry.sourceAffineLocally_morphismRestrict, AlgebraicGeometry.Scheme.comp_appLE_assoc, AlgebraicGeometry.Proj.awayToSection_comp_appLE_assoc, AlgebraicGeometry.flat_iff, AlgebraicGeometry.isIso_pushoutSection_of_isAffineOpen, AlgebraicGeometry.IsAffineOpen.appLE_eq_away_map, appIso_inv_appLE, AlgebraicGeometry.IsOpenImmersion.map_ΓIso_inv_apply, AlgebraicGeometry.Smooth.iff_forall_exists_isStandardSmooth
appTop 📖	CompOp	74 math math: AlgebraicGeometry.Γ_map_morphismRestrict, AlgebraicGeometry.Scheme.ΓSpecIso_inv_naturality_assoc, AlgebraicGeometry.IsAffineOpen.isoSpec_inv_appTop, RingHom.IsStableUnderBaseChange.pullback_fst_appTop, AlgebraicGeometry.AffineSpace.SpecIso_hom_appTop, id_appTop, AlgebraicGeometry.Scheme.ideal_ker_le_ker_ΓSpecIso_inv_comp, AlgebraicGeometry.Scheme.toSpecΓ_naturality_assoc, AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv_appTop_coord, AlgebraicGeometry.Scheme.isoSpec_hom_naturality, AlgebraicGeometry.Scheme.toSpecΓ_naturality, AlgebraicGeometry.Scheme.preimage_basicOpen_top, AlgebraicGeometry.Scheme.isoSpec_hom_naturality_assoc, AlgebraicGeometry.Scheme.fromSpecStalk_appTop, AlgebraicGeometry.Scheme.id_appTop, AlgebraicGeometry.Scheme.Opens.toSpecΓ_appTop_assoc, AlgebraicGeometry.Scheme.ker_of_isAffine, AlgebraicGeometry.Scheme.Γevaluation_naturality_apply, AlgebraicGeometry.Flat.flat_and_surjective_iff_faithfullyFlat_of_isAffine, AlgebraicGeometry.isIntegral_appTop_of_universallyClosed, AlgebraicGeometry.Scheme.Opens.toSpecΓ_appTop, AlgebraicGeometry.ΓSpecIso_obj_hom, preimage_basicOpen_top, AlgebraicGeometry.exists_appTop_π_eq_of_isAffine_of_isLimit, AlgebraicGeometry.Scheme.Γevaluation_naturality_assoc, AlgebraicGeometry.AffineSpace.SpecIso_inv_appTop_coord, AlgebraicGeometry.AffineSpace.comp_homOfVector, AlgebraicGeometry.Scheme.homOfLE_appTop, AlgebraicGeometry.Scheme.isoSpec_inv_naturality_assoc, AlgebraicGeometry.AffineSpace.hom_ext_iff, AlgebraicGeometry.exists_appTop_map_eq_zero_of_isAffine_of_isLimit, AlgebraicGeometry.HasRingHomProperty.appTop, AlgebraicGeometry.Scheme.isPullback_toSpecΓ_toSpecΓ, AlgebraicGeometry.Scheme.ΓSpecIso_naturality, AlgebraicGeometry.Scheme.Γ_map_op, AlgebraicGeometry.IsFinite.instHasAffinePropertyAndIsAffineFiniteCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTopHomAppTop, AlgebraicGeometry.Scheme.toSpecΓ_appTop, AlgebraicGeometry.Scheme.Γ_map, AlgebraicGeometry.Scheme.Opens.ι_appTop, AlgebraicGeometry.Scheme.isoSpec_inv_naturality, AlgebraicGeometry.Scheme.comp_appTop_assoc, AlgebraicGeometry.IsClosedImmersion.hasAffineProperty, AlgebraicGeometry.Scheme.preimage_opensRange_toSpecΓ, AlgebraicGeometry.Scheme.inv_appTop, AlgebraicGeometry.morphismRestrict_appTop, AlgebraicGeometry.AffineSpace.toSpecMvPolyIntEquiv_apply, AlgebraicGeometry.ΓSpecIso_inv_ΓSpec_adjunction_homEquiv, inv_appTop, AlgebraicGeometry.Scheme.ΓSpecIso_inv_naturality, AlgebraicGeometry.AffineSpace.isoOfIsAffine_hom_appTop, AlgebraicGeometry.finite_appTop_of_universallyClosed, AlgebraicGeometry.specTargetImageFactorization_app_injective, AlgebraicGeometry.AffineSpace.isoOfIsAffine_hom, AlgebraicGeometry.affineAnd_apply, AlgebraicGeometry.AffineSpace.reindex_appTop_coord, AlgebraicGeometry.exists_appTop_π_eq_of_isLimit, AlgebraicGeometry.AffineSpace.homOverEquiv_apply, AlgebraicGeometry.Scheme.comp_appTop, AlgebraicGeometry.IsClosedImmersion.isAffine_surjective_of_isAffine, AlgebraicGeometry.AffineSpace.toSpecMvPolyIntEquiv_comp, comp_appTop, AlgebraicGeometry.instHasAffinePropertyIsomorphismsSchemeAndIsAffineIsIsoCommRingCatAppTop, AlgebraicGeometry.exists_appTop_map_eq_zero_of_isLimit, AlgebraicGeometry.Scheme.ΓSpecIso_naturality_assoc, AlgebraicGeometry.ΓSpec_adjunction_homEquiv_eq, AlgebraicGeometry.Scheme.Γevaluation_naturality, AlgebraicGeometry.Scheme.IdealSheafData.ker_glueDataObjι_appTop, AlgebraicGeometry.AffineSpace.comp_homOfVector_assoc, AlgebraicGeometry.HasRingHomProperty.iff_of_source_openCover, comp_appTop_assoc, AlgebraicGeometry.AffineSpace.map_appTop_coord, AlgebraicGeometry.AffineSpace.homOfVector_appTop_coord, AlgebraicGeometry.IsAffineOpen.isoSpec_hom_appTop, AlgebraicGeometry.HasRingHomProperty.iff_of_isAffine
arrowStalkMapIsoOfEq 📖	CompOp	—
copyBase 📖	CompOp	1 math math: copyBase_eq
homeomorph 📖	CompOp	1 math math: homeomorph_apply
stalkMap 📖	CompOp	76 math math: germ_stalkMap_assoc, AlgebraicGeometry.Scheme.Spec_map_stalkMap_fromSpecStalk, AlgebraicGeometry.surjectiveOnStalks_iff, stalkMap_congr_point, AlgebraicGeometry.HasRingHomProperty.stalkMap, AlgebraicGeometry.Scheme.stalkSpecializes_stalkMap, AlgebraicGeometry.Scheme.stalkMap_congr_assoc, stalkSpecializes_stalkMap, AlgebraicGeometry.Scheme.Opens.stalkIso_inv, stalkMap_congr_hom_assoc, AlgebraicGeometry.IsOpenImmersion.iff_isIso_stalkMap, AlgebraicGeometry.Scheme.Cover.instIsIsoCommRingCatStalkMapFromGlued, AlgebraicGeometry.isIso_iff_isIso_stalkMap, AlgebraicGeometry.IsAffineOpen.stalkMap_injective, germ_stalkMap_apply, stalkMap_congr_point_assoc, AlgebraicGeometry.Scheme.stalkMap_congr_point_assoc, AlgebraicGeometry.Scheme.Spec_map_stalkMap_fromSpecStalk_assoc, AlgebraicGeometry.Scheme.stalkMap_hom_inv_assoc, AlgebraicGeometry.formallySmooth_stalkMap_iff, AlgebraicGeometry.isIso_stalkMap_coprodSpec, AlgebraicGeometry.Scheme.stalkMap_germ_assoc, AlgebraicGeometry.Scheme.stalkMap_congr_hom_assoc, AlgebraicGeometry.Scheme.stalkSpecializes_stalkMap_apply, AlgebraicGeometry.Scheme.stalkMap_id, AlgebraicGeometry.Scheme.SpecMap_stalkMap_fromSpecStalk_assoc, AlgebraicGeometry.isOpenImmersion_iff_stalk, AlgebraicGeometry.SurjectiveOnStalks.surj_on_stalks, stalkSpecializes_stalkMap_assoc, AlgebraicGeometry.SurjectiveOnStalks.stalkMap_surjective, stalkMap_hom_inv, AlgebraicGeometry.Scheme.stalkMap_congr_hom, AlgebraicGeometry.Scheme.stalkMap_comp, stalkMap_inv_hom_assoc, AlgebraicGeometry.LocallyOfFiniteType.stalkMap, AlgebraicGeometry.Scheme.stalkMap_congr_point, AlgebraicGeometry.Scheme.stalkMap_hom_inv, AlgebraicGeometry.Scheme.stalkMap_hom_inv_apply, stalkSpecializes_stalkMap_apply, AlgebraicGeometry.IsOpenImmersion.instIsIsoCommRingCatStalkMap, AlgebraicGeometry.stalkMap_injective_of_isOpenMap_of_injective, AlgebraicGeometry.Scheme.stalkMap_germ_apply, germ_stalkMap, AlgebraicGeometry.stalkMap_injective_of_isAffine, AlgebraicGeometry.Flat.stalkMap, stalkMap_comp, AlgebraicGeometry.Scheme.residue_residueFieldMap_assoc, AlgebraicGeometry.FormallyUnramified.stalkMap, AlgebraicGeometry.IsOpenImmersion.iff_stalk_iso, AlgebraicGeometry.Scheme.residue_residueFieldMap, AlgebraicGeometry.Scheme.stalkMap_inv_hom_apply, mem_smoothLocus, stalkMap_hom_inv_apply, AlgebraicGeometry.Scheme.stalkMap_inv_hom, AlgebraicGeometry.Flat.iff_flat_stalkMap, AlgebraicGeometry.isIso_iff_stalk_iso, AlgebraicGeometry.Scheme.instIsOverMapStalkMapOverInferInstanceOverClass, stalkMap_congr, stalkMap_congr_assoc, AlgebraicGeometry.Scheme.stalkMap_inv_hom_assoc, AlgebraicGeometry.HasRingHomProperty.stalkMap_of_respectsIso, stalkMap_id, AlgebraicGeometry.Scheme.SpecMap_stalkMap_fromSpecStalk, stalkMap_inv_hom_apply, stalkMap_hom_inv_assoc, instIsLocalHomCarrierStalkCommRingCatPresheafCoeContinuousMapCarrierCarrierHomTopCatBaseRingHomHomStalkMap, AlgebraicGeometry.Scheme.stalkClosedPointTo_comp, stalkMap_inv_hom, AlgebraicGeometry.Scheme.localRingHom_comp_stalkIso, AlgebraicGeometry.Scheme.stalkMap_germ, AlgebraicGeometry.Scheme.localRingHom_comp_stalkIso_apply, isOpen_smoothLocus, AlgebraicGeometry.Scheme.stalkSpecializes_stalkMap_assoc, AlgebraicGeometry.Scheme.stalkMap_congr, stalkMap_congr_hom, stalkMap_surjective
toLRSHom 📖	CompOp	24 math math: AlgebraicGeometry.Scheme.comp_toLRSHom_assoc, AlgebraicGeometry.Scheme.ofRestrict_toLRSHom_base, AlgebraicGeometry.Scheme.forgetToTop_map, AlgebraicGeometry.Scheme.forgetToLocallyRingedSpace_map, AlgebraicGeometry.Scheme.Opens.germ_stalkIso_inv, AlgebraicGeometry.morphismRestrict_appLE, AlgebraicGeometry.Scheme.Opens.germ_stalkIso_inv_assoc, isIso_toLRSHom, AlgebraicGeometry.Scheme.Opens.germ_stalkIso_hom, id_appIso, AlgebraicGeometry.Scheme.ofRestrict_toLRSHom_c_app, AlgebraicGeometry.Scheme.fullyFaithfulForgetToLocallyRingedSpace_preimage_toLRSHom, comp_toLRSHom, AlgebraicGeometry.Scheme.Opens.germ_stalkIso_hom_assoc, comp_toLRSHom_assoc, AlgebraicGeometry.Scheme.comp_toLRSHom, AlgebraicGeometry.ΓSpec.adjunction_homEquiv_symm_apply, AlgebraicGeometry.Scheme.GlueData.ι_isoLocallyRingedSpace_inv, AlgebraicGeometry.IsAffineOpen.isLocalization_stalk', AlgebraicGeometry.Scheme.Opens.ι_appLE, opensRange_comp_of_isIso, inv_image, opensRange_of_isIso, id_image
toLRSHom' 📖	CompOp	656 math math: AlgebraicGeometry.Scheme.toSpecΓ_apply, resLE_map_assoc, AlgebraicGeometry.Γ_map_morphismRestrict, AlgebraicGeometry.Scheme.Modules.pushforward_obj_presheaf_map, AlgebraicGeometry.QuasiCompact.isCompact_preimage_singleton, AlgebraicGeometry.Scheme.fromSpecStalk_closedPoint, AlgebraicGeometry.Scheme.coprodPresheafObjIso_hom_fst_assoc, AlgebraicGeometry.Proj.fromOfGlobalSections_resLE, AlgebraicGeometry.Scheme.map_PrimeSpectrum_basicOpen_of_affine, AlgebraicGeometry.Scheme.germ_stalkClosedPointTo, opensRange_pullbackSnd, germ_stalkMap_assoc, AlgebraicGeometry.Scheme.Spec_map_stalkMap_fromSpecStalk, image_preimage_eq_opensRange_inter, AlgebraicGeometry.Scheme.Cover.LocallyDirected.directed, AlgebraicGeometry.QuasiCompact.isCompact_preimage, AlgebraicGeometry.Scheme.Pullback.Triplet.mk'_s, app_invApp'_assoc, denseRange, AlgebraicGeometry.Scheme.isEmpty_pullback_iff, quasiFiniteLocus_comp, AlgebraicGeometry.Scheme.app_eq, closePoints_subset_preimage_closedPoints, AlgebraicGeometry.opensCone_pt, AlgebraicGeometry.coprodSpec_apply, AlgebraicGeometry.morphismRestrict_base_coe, fiberHomeo_apply, AlgebraicGeometry.surjectiveOnStalks_iff, AlgebraicGeometry.Scheme.germ_stalkClosedPointTo_Spec_fromSpecStalk_assoc, AlgebraicGeometry.Scheme.affineBasisCover_is_basis, AlgebraicGeometry.Scheme.Opens.toSpecΓ_naturality, AlgebraicGeometry.range_eq_range_of_surjective, AlgebraicGeometry.Scheme.Cover.isOpenMap_fromGlued, AlgebraicGeometry.Scheme.Modules.restrictAdjunction_counit_app_app, AlgebraicGeometry.instGeometricallyIrreducibleMorphismRestrict, AlgebraicGeometry.Scheme.Pullback.Triplet.ofPoint_x, exists_isIso_morphismRestrict_toNormalization, AlgebraicGeometry.morphismRestrict_ι_assoc, stalkMap_congr_point, AlgebraicGeometry.ValuativeCriterion.Existence.specializingMap, AlgebraicGeometry.Scheme.Modules.pushforwardCongr_hom_app_app, AlgebraicGeometry.exists_isAffineOpen_preimage_eq, AlgebraicGeometry.Scheme.Modules.pushforward_obj_obj, range_fiberι, preimage_bot, AlgebraicGeometry.Scheme.Cover.iUnion_range, AlgebraicGeometry.pullbackRestrictIsoRestrict_hom_morphismRestrict_assoc, mem_opensRange, AlgebraicGeometry.isCompact_iff_exists, AlgebraicGeometry.IsAffineOpen.isCompact_pullback_inf, AlgebraicGeometry.HasRingHomProperty.stalkMap, resLE_eq_morphismRestrict, AlgebraicGeometry.Scheme.inv_app, AlgebraicGeometry.image_morphismRestrict_preimage, AlgebraicGeometry.Scheme.emptyTo_base, AlgebraicGeometry.Proj.map_preimage_basicOpen, AlgebraicGeometry.IsOpenImmersion.comp_lift_assoc, AlgebraicGeometry.Scheme.OpenCover.finiteSubcover_f, AlgebraicGeometry.Scheme.stalkSpecializes_stalkMap, AlgebraicGeometry.Scheme.Modules.germ_restrictStalkNatIso_inv_app, AlgebraicGeometry.SpecMap_preimage_basicOpen, AlgebraicGeometry.Scheme.stalkMap_congr_assoc, AlgebraicGeometry.instIsGermInjectiveAtCoeContinuousMapCarrierCarrierCommRingCatHomTopCatBaseOfIsOpenImmersion, AlgebraicGeometry.Scheme.Pullback.SpecTensorTo_SpecOfPoint, AlgebraicGeometry.Scheme.hom_inv_apply, range_asFiberHom, AlgebraicGeometry.isIso_ΓSpec_adjunction_unit_app_basicOpen, tendsto_cofinite_cofinite, preimage_le_preimage_of_le, stalkSpecializes_stalkMap, coe_resLE_base, AlgebraicGeometry.IsAffineOpen.fromSpecStalk_closedPoint, AlgebraicGeometry.SmoothOfRelativeDimension.exists_isStandardSmoothOfRelativeDimension, preimageIso_hom_ι_assoc, AlgebraicGeometry.IsClosedImmersion.base_closed, AlgebraicGeometry.quasiCompact_iff, AlgebraicGeometry.isPullback_fiberToSpecResidueField_of_isPullback, AlgebraicGeometry.Scheme.congr_app, AlgebraicGeometry.IsZariskiLocalAtTarget.restrict, AlgebraicGeometry.Scheme.preimage_comp, stalkMap_congr_hom_assoc, AlgebraicGeometry.Scheme.Cover.RelativeGluingData.toBase_preimage_eq_opensRange_ι, AlgebraicGeometry.IsFinite.instMorphismRestrict, AlgebraicGeometry.Scheme.Pullback.Triplet.isPullback_SpecMap_tensor, AlgebraicGeometry.Scheme.forget_map, AlgebraicGeometry.sigmaMk_mk, AlgebraicGeometry.Scheme.Pullback.Triplet.specTensorTo_base_snd, AlgebraicGeometry.Scheme.isoSpec_inv_preimage_zeroLocus, AlgebraicGeometry.effectiveEpi_base_of_flat, AlgebraicGeometry.IsPreimmersion.isEmbedding, irreducibleComponentsEquiv_symm_apply_coe, SpecMap_residueFieldMap_fromSpecResidueField_assoc, AlgebraicGeometry.IsOpenImmersion.iff_isIso_stalkMap, AlgebraicGeometry.Scheme.Cover.instIsIsoCommRingCatStalkMapFromGlued, preimage_top, AlgebraicGeometry.Scheme.coprodPresheafObjIso_hom_snd_assoc, toNormalization_app_preimage, AlgebraicGeometry.Scheme.Pullback.carrierEquiv_eq_iff, toImage_app_injective, id_preimage, AlgebraicGeometry.Flat.instMorphismRestrict, AlgebraicGeometry.Scheme.germ_stalkClosedPointTo_Spec_fromSpecStalk, AlgebraicGeometry.Scheme.IdealSheafData.ideal_le_ker_glueDataObjι, AlgebraicGeometry.SpecToEquivOfLocalRing_apply_snd_coe, AlgebraicGeometry.instQuasiCompactMorphismRestrict, AlgebraicGeometry.Scheme.eqToHom_c_app, AlgebraicGeometry.Scheme.exists_affine_mem_range_and_range_subset, image_preimage_le, AlgebraicGeometry.IsOpenImmersion.isOpen_range, AlgebraicGeometry.Scheme.affineOpenCover_idx, AlgebraicGeometry.Scheme.Opens.isoOfLE_hom_ι_assoc, fromNormalization_preimage, iSup_preimage_eq_top, AlgebraicGeometry.Scheme.Pullback.Triplet.snd_SpecTensorTo_apply, AlgebraicGeometry.SurjectiveOnStalks.iff_of_isAffine, AlgebraicGeometry.IsIntegralHom.isIntegral_app, AlgebraicGeometry.Scheme.presieve₀_mem_precoverage_iff, AlgebraicGeometry.Scheme.comp_base, image_preimage_eq_opensRange_inf, AlgebraicGeometry.isIso_iff_isIso_stalkMap, AlgebraicGeometry.IsAffineOpen.stalkMap_injective, appLE_map', AlgebraicGeometry.Scheme.Pullback.residueFieldCongr_inv_residueFieldMap_ofPoint, AlgebraicGeometry.Scheme.IdealSheafData.ideal_map, AlgebraicGeometry.IsOpenImmersion.range_pullbackSnd, AlgebraicGeometry.IsImmersion.instMorphismRestrict, resLE_map, isDiscrete_preimage_singleton, comp_app, AlgebraicGeometry.Scheme.IdealSheafData.range_glueDataObjι_ι_eq_support_inter, AlgebraicGeometry.Scheme.hom_base_inv_base, AlgebraicGeometry.continuousMapPresheaf_map, AlgebraicGeometry.IsOpenImmersion.range_pullback_fst_of_right, AlgebraicGeometry.Proj.ι_comp_map, coe_resLE_apply, AlgebraicGeometry.pointEquivClosedPoint_apply_coe, ι_toNormalization, AlgebraicGeometry.Scheme.range_fromSpecStalk, AlgebraicGeometry.Scheme.comp_coeBase_assoc, AlgebraicGeometry.Scheme.comp_appLE, congr_app, preimage_iSup, AlgebraicGeometry.coprodMk_inl, AlgebraicGeometry.Scheme.Opens.toSpecΓ_naturality_assoc, AlgebraicGeometry.Scheme.Pullback.Triplet.hy, AlgebraicGeometry.Scheme.Pullback.carrierEquiv_symm_fst, asFiberHom_fiberι_assoc, germ_stalkMap_apply, AlgebraicGeometry.Scheme.preimage_basicOpen_top, app_appIso_inv_assoc, AlgebraicGeometry.IsAffineOpen.basicOpen_fromSpec_app, AlgebraicGeometry.Scheme.opensRange_homOfLE, AlgebraicGeometry.Scheme.Pullback.range_fst_comp, AlgebraicGeometry.Scheme.isoSpec_inv_image_zeroLocus, AlgebraicGeometry.Scheme.IdealSheafData.range_glueDataObjι_ι, isIso_base, AlgebraicGeometry.Scheme.Modules.germ_restrictStalkNatIso_hom_app, AlgebraicGeometry.isIso_morphismRestrict_iff_isIso_app, AlgebraicGeometry.Scheme.hom_base_inv_base_assoc, AlgebraicGeometry.morphismRestrict_base, AlgebraicGeometry.Scheme.ι_image_homOfLE_le_ι_image, AlgebraicGeometry.Scheme.toSpecΓ_preimage_zeroLocus, preimage_image_eq, quasiFiniteAt_comp_iff_of_isOpenImmersion, AlgebraicGeometry.isBasis_preimage_isAffineOpen, AlgebraicGeometry.Scheme.Cover.isOpenEmbedding_fromGlued, app_surjective, AlgebraicGeometry.coe_opensRestrict_symm_apply, AlgebraicGeometry.pullbackRestrictIsoRestrict_inv_fst, isCompact_preimage_singleton, AlgebraicGeometry.Scheme.IdealSheafData.vanishingIdeal_ideal, AlgebraicGeometry.Scheme.Pullback.Triplet.Spec_map_tensor_isPullback, AlgebraicGeometry.IsLocalAtSource.iff_exists_resLE, AlgebraicGeometry.Scheme.Cover.covers, AlgebraicGeometry.Scheme.Spec_map_presheaf_map_eqToHom, stalkMap_congr_point_assoc, AlgebraicGeometry.Scheme.stalkMap_congr_point_assoc, AlgebraicGeometry.Scheme.GlueData.isOpen_iff, AlgebraicGeometry.QuasiCompactCover.isCompactOpenCovered_of_isCompact, mem_preimage, AlgebraicGeometry.Scheme.Spec_map_stalkMap_fromSpecStalk_assoc, isConstructible_preimage, AlgebraicGeometry.IsImmersion.isLocallyClosed_range, AlgebraicGeometry.Scheme.Γevaluation_naturality_apply, AlgebraicGeometry.ExistsHomHomCompEqCompAux.exists_index, AlgebraicGeometry.liftCoborder_app, AlgebraicGeometry.Scheme.Opens.toSpecΓ_preimage_basicOpen, AlgebraicGeometry.Scheme.Cover.exists_lift_trans_eq, AlgebraicGeometry.Scheme.Cover.exists_of_f_eq_f, AlgebraicGeometry.Scheme.stalkClosedPointTo_fromSpecStalk, AlgebraicGeometry.IsOpenImmersion.map_ΓIso_inv_assoc, preimage_smoothLocus_eq, AlgebraicGeometry.Scheme.coe_homeoOfIso_symm, AlgebraicGeometry.Scheme.stalkMap_hom_inv_assoc, AlgebraicGeometry.Scheme.id.base, AlgebraicGeometry.Scheme.IdealSheafData.range_subschemeι, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.scheme_toScheme, isClosedMap, isLocallyClosed_range, AlgebraicGeometry.opensDiagram_map, AlgebraicGeometry.morphismRestrict_appLE, appIso_inv_app_presheafMap, AlgebraicGeometry.IsClosedImmersion.iff_isPreimmersion, app_eq, AlgebraicGeometry.morphismRestrict_ι, comp_appLE, preimageIso_inv_ι, apply_mem_image_iff, asFiberHom_apply, AlgebraicGeometry.genericPoint_eq_of_isOpenImmersion, AlgebraicGeometry.Scheme.homeoOfIso_apply, preimage_basicOpen_top, AlgebraicGeometry.formallySmooth_stalkMap_iff, AlgebraicGeometry.isIso_stalkMap_coprodSpec, AlgebraicGeometry.WeaklyEtale.instMorphismRestrict, AlgebraicGeometry.Scheme.instIsIsoCommRingCatResidueFieldMapOfIsOpenImmersion, AlgebraicGeometry.Scheme.instIsOverMapResidueFieldMapOverInferInstanceOverClass, AlgebraicGeometry.HasAffineProperty.restrict, AlgebraicGeometry.Scheme.Γevaluation_naturality_assoc, AlgebraicGeometry.opensDiagramι_app, AlgebraicGeometry.Scheme.Pullback.carrierEquiv_symm_snd, AlgebraicGeometry.isCompl_range_inl_inr, ker_apply, AlgebraicGeometry.Scheme.Pullback.range_map, AlgebraicGeometry.SurjectiveOnStalks.isEmbedding_pullback, isSpectralMap, AlgebraicGeometry.mem_range_iff_of_surjective, AlgebraicGeometry.instIsIsoSchemeMorphismRestrict, isLocallyConstructible_image, AlgebraicGeometry.Scheme.stalkMap_germ_assoc, appIso_inv_app, liftCoborder_preimage, AlgebraicGeometry.Scheme.stalkMap_congr_hom_assoc, opensRange_pullbackFst, AlgebraicGeometry.Scheme.stalkSpecializes_stalkMap_apply, AlgebraicGeometry.pullbackRestrictIsoRestrict_hom_ι_assoc, AlgebraicGeometry.exists_map_preimage_eq_map_preimage, map_appLE_assoc, AlgebraicGeometry.Scheme.comp_base_assoc, AlgebraicGeometry.instLocallyQuasiFiniteMorphismRestrict, map_appLE'_assoc, comp_app_assoc, AlgebraicGeometry.Scheme.comp_app, finite_app, AlgebraicGeometry.isIso_iff_isOpenImmersion, AlgebraicGeometry.instGeometricallyReducedMorphismRestrict, AlgebraicGeometry.IsAffineOpen.fromSpec_preimage_zeroLocus, AlgebraicGeometry.exists_preimage_eq, AlgebraicGeometry.Scheme.evaluation_naturality, AlgebraicGeometry.Scheme.GlueData.ι_eq_iff, AlgebraicGeometry.FormallyUnramified.instIsSeparableCarrierResidueFieldCoeContinuousMapCarrierCarrierCommRingCatHomTopCatBaseOfLocallyOfFiniteType, ideal_ker_le, appIso_hom', AlgebraicGeometry.AffineSpace.isOpenMap_over, AlgebraicGeometry.surjective_iff, AlgebraicGeometry.Scheme.homOfLE_appTop, AlgebraicGeometry.IsAffineOpen.fromSpec_app_self, AlgebraicGeometry.isClosedImmersion_iff_isAffineHom, resLE_congr, AlgebraicGeometry.Scheme.SpecMap_stalkMap_fromSpecStalk_assoc, preimage_basicOpen, AlgebraicGeometry.Scheme.singleton_mem_precoverage_iff, AlgebraicGeometry.Scheme.IdealSheafData.subschemeι_app, appLE_appIso_inv, isOpenMap, AlgebraicGeometry.Scheme.Pullback.Triplet.hx, app_invApp', stalkFunctor_toImage_injective, AlgebraicGeometry.isOpenImmersion_iff_stalk, AlgebraicGeometry.SurjectiveOnStalks.surj_on_stalks, stalkSpecializes_stalkMap_assoc, AlgebraicGeometry.SurjectiveOnStalks.stalkMap_surjective, AlgebraicGeometry.morphismRestrict_comp, comp_appLE_assoc, AlgebraicGeometry.IsProper.instMorphismRestrict, AlgebraicGeometry.IsZariskiLocalAtTarget.iff_of_iSup_eq_top, stalkMap_hom_inv, AlgebraicGeometry.Scheme.IdealSheafData.support_comap, AlgebraicGeometry.Scheme.stalkMap_congr_hom, AlgebraicGeometry.Scheme.Pullback.range_snd_comp, AlgebraicGeometry.Scheme.Opens.isoOfLE_hom_ι, AlgebraicGeometry.Scheme.iso_inv_base_hom_base, AlgebraicGeometry.range_eq_univ, AlgebraicGeometry.SpecToEquivOfLocalRing_apply_fst, comp_apply, isOpenEmbedding, AlgebraicGeometry.opensDiagram_obj, AlgebraicGeometry.Spec.map_apply, AlgebraicGeometry.pointOfClosedPoint_apply, AlgebraicGeometry.morphismRestrict_app', AlgebraicGeometry.Scheme.evaluation_naturality_apply, isEmbedding, AlgebraicGeometry.sigmaι_eq_iff, AlgebraicGeometry.opensCone_π_app, appLE_map_assoc, AlgebraicGeometry.IsOpenImmersion.opensEquiv_symm_apply, AlgebraicGeometry.Scheme.irreducibleComponentι_apply, AlgebraicGeometry.Scheme.stalkMap_comp, AlgebraicGeometry.IsOpenImmersion.range_pullback_to_base_of_left, AlgebraicGeometry.Scheme.IdealSheafData.map_vanishingIdeal, stalkMap_inv_hom_assoc, AlgebraicGeometry.targetAffineLocally_affineAnd_iff, appIso_inv_app_assoc, AlgebraicGeometry.IsAffineOpen.fromSpec_preimage_self, residueFieldMap_congr, map_mem_image_iff, AlgebraicGeometry.Scheme.coprodPresheafObjIso_hom_snd, AlgebraicGeometry.coprodSpec_coprodMk, AlgebraicGeometry.Scheme.Cover.exists_of_trans_eq_trans, AlgebraicGeometry.Scheme.IdealSheafData.opensRange_subschemeCover_map, preimageIso_hom_ι, AlgebraicGeometry.exists_affineOpens_le_appLE_of_appLE, AlgebraicGeometry.affinePreimage, injective, AlgebraicGeometry.isIso_iff_isOpenImmersion_and_epi_base, preimageIso_inv_ι_assoc, AlgebraicGeometry.isAffineHom_iff, asFiberHom_fiberι, AlgebraicGeometry.exists_smooth_of_formallySmooth_stalk, resLE_comp_resLE_assoc, AlgebraicGeometry.LocallyOfFiniteType.stalkMap, AlgebraicGeometry.locallyQuasiFinite_iff_isDiscrete_preimage_singleton, AlgebraicGeometry.Scheme.IsLocallyDirected.ι_eq_ι_iff, AlgebraicGeometry.instLocallyOfFiniteTypeMorphismRestrict, appLE_congr, AlgebraicGeometry.IsIntegralHom.instMorphismRestrict, AlgebraicGeometry.Scheme.Pullback.Triplet.specTensorTo_base_fst, AlgebraicGeometry.compactSpace_iff_exists, preimage_sup, AlgebraicGeometry.Scheme.preimage_opensRange_toSpecΓ, AlgebraicGeometry.instIsOpenImmersionMorphismRestrict, AlgebraicGeometry.Scheme.iso_hom_base_inv_base, AlgebraicGeometry.Scheme.stalkMap_congr_point, AlgebraicGeometry.Scheme.Opens.ι_preimage_self, AlgebraicGeometry.IsAffineOpen.comap_primeIdealOf_appLE, AlgebraicGeometry.Scheme.stalkMap_hom_inv, appLE_appIso_inv_assoc, AlgebraicGeometry.Scheme.toSpecΓ_image_zeroLocus, AlgebraicGeometry.instQuasiSeparatedMorphismRestrict, AlgebraicGeometry.HasAffineProperty.iff_of_iSup_eq_top, AlgebraicGeometry.isImmersion_iff, AlgebraicGeometry.Scheme.homOfLE_app, comp_preimage, naturality_assoc, AlgebraicGeometry.Scheme.IdealSheafData.opensRange_glueDataObjMap, AlgebraicGeometry.IsAffineOpen.fromSpec_app_self_apply, AlgebraicGeometry.Scheme.stalkMap_hom_inv_apply, AlgebraicGeometry.Scheme.homOfLE_appLE, isIso_app, AlgebraicGeometry.IsAffineOpen.ι_basicOpen_preimage, AlgebraicGeometry.Scheme.basicOpen_appLE, stalkSpecializes_stalkMap_apply, AlgebraicGeometry.morphismRestrict_appTop, exists_mem_and_isIso_morphismRestrict_toNormalization, AlgebraicGeometry.isOpenMap_of_generalizingMap, preimage_opensRange, AlgebraicGeometry.Scheme.descResidueField_stalkClosedPointTo_fromSpecResidueField, AlgebraicGeometry.IsAffineOpen.fromSpec_preimage_basicOpen, AlgebraicGeometry.isDominant_iff, AlgebraicGeometry.Flat.generalizingMap, fiberι_asFiber, AlgebraicGeometry.HasAffineProperty.affineAnd_iff, AlgebraicGeometry.IsOpenImmersion.app_eq_appIso_inv_app_of_comp_eq, id_base, AlgebraicGeometry.Scheme.IdealSheafData.ker_subschemeι_app, AlgebraicGeometry.IsOpenImmersion.ΓIso_inv, AlgebraicGeometry.IsOpenImmersion.range_pullback_snd_of_left, AlgebraicGeometry.IsOpenImmersion.instIsIsoCommRingCatStalkMap, AlgebraicGeometry.isPreimmersion_iff, AlgebraicGeometry.Scheme.AffineCover.covers, AlgebraicGeometry.Scheme.Opens.isoOfLE_inv_ι, ι_toNormalization_assoc, Spec_map_residueFieldMap_fromSpecResidueField_assoc, AlgebraicGeometry.Scheme.isLocalHom_stalkClosedPointTo', AlgebraicGeometry.IsFinite.finite_preimage_singleton, AlgebraicGeometry.IsLocalAtTarget.restrict, fromNormalization_app_assoc, AlgebraicGeometry.Proj.fromOfGlobalSections_morphismRestrict, AlgebraicGeometry.IsAffineOpen.primeIdealOf_eq_map_closedPoint, AlgebraicGeometry.stalkMap_injective_of_isOpenMap_of_injective, AlgebraicGeometry.Scheme.GlueData.ι_isoCarrier_inv, AlgebraicGeometry.Scheme.stalkMap_germ_apply, AlgebraicGeometry.Scheme.IdealSheafData.range_glueDataObjι, instIsIsoCommRingCatAppObjOpensOpensFunctor, AlgebraicGeometry.Scheme.ker_morphismRestrict_ideal, AlgebraicGeometry.pullbackRestrictIsoRestrict_hom_ι, germ_stalkMap, AlgebraicGeometry.Scheme.coprodPresheafObjIso_hom_fst, AlgebraicGeometry.IsOpenImmersion.range_pullback_to_base_of_right, AlgebraicGeometry.IsOpenImmersion.app_ΓIso_hom_assoc, preimage_iSup_eq_top, AlgebraicGeometry.Scheme.Modules.restrictAdjunction_unit_app_app, AlgebraicGeometry.IsLocalAtTarget.iff_of_iSup_eq_top, isIntegral_app, app_eq_appLE, AlgebraicGeometry.Scheme.Pullback.Triplet.fst_SpecTensorTo_apply, AlgebraicGeometry.IsOpenImmersion.map_ΓIso_inv, AlgebraicGeometry.exists_map_preimage_le_map_preimage, AlgebraicGeometry.IsClosedImmersion.isClosedEmbedding, map_appLE', AlgebraicGeometry.isIntegralHom_iff, AlgebraicGeometry.stalkMap_injective_of_isAffine, AlgebraicGeometry.Scheme.IsJointlySurjectivePreserving.exists_preimage_snd_triplet_of_prop, AlgebraicGeometry.exists_app_map_eq_map_of_isLimit, AlgebraicGeometry.Flat.stalkMap, AlgebraicGeometry.Scheme.Pullback.range_snd, stalkMap_comp, AlgebraicGeometry.Scheme.Cover.exists_eq, AlgebraicGeometry.IsOpenImmersion.app_ΓIso_hom, AlgebraicGeometry.Scheme.ofArrows_mem_precoverage_iff, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toSchemeHom_toPshHom, AlgebraicGeometry.Scheme.IsLocallyDirected.ι_jointly_surjective, AlgebraicGeometry.IsPreimmersion.base_embedding, coe_image, coe_preimage, AlgebraicGeometry.Scheme.preimage_eq_top_of_closedPoint_mem, isClosedEmbedding, AlgebraicGeometry.Proj.awayι_preimage_basicOpen, quasiFiniteAt_iff_isOpen_singleton_asFiber, AlgebraicGeometry.IsFinite.finite_app, AlgebraicGeometry.Scheme.germ_stalkClosedPointTo_assoc, finite_preimage_singleton, map_resLE, AlgebraicGeometry.Scheme.inv_hom_apply, AlgebraicGeometry.exists_mem_of_isClosed_of_nonempty', AlgebraicGeometry.IsIntegralHom.integral_app, resLE_preimage, AlgebraicGeometry.Scheme.residue_residueFieldMap_assoc, AlgebraicGeometry.Scheme.IdealSheafData.support_map, AlgebraicGeometry.FormallyUnramified.stalkMap, le_preimage_resLE_iff, AlgebraicGeometry.IsAffineOpen.preimage_of_isIso, AlgebraicGeometry.instIsClosedImmersionMorphismRestrict, preimage_inf, AlgebraicGeometry.isPullback_morphismRestrict, appIso_hom, AlgebraicGeometry.Scheme.OpenCover.finiteSubcover_X, AlgebraicGeometry.Scheme.Pullback.range_fst, AlgebraicGeometry.IsOpenImmersion.iff_stalk_iso, AlgebraicGeometry.Scheme.Cover.fromGlued_isOpenEmbedding, AlgebraicGeometry.IsOpenImmersion.app_ΓIso_hom_apply, AlgebraicGeometry.Scheme.residue_residueFieldMap, AlgebraicGeometry.Scheme.evaluation_naturality_assoc, AlgebraicGeometry.Scheme.emptyTo_c_app, AlgebraicGeometry.Scheme.stalkMap_inv_hom_apply, map_resLE_assoc, comp_base_assoc, AlgebraicGeometry.IsAffineOpen.preimage_of_isOpenImmersion, AlgebraicGeometry.Scheme.SpecMap_residue_apply, AlgebraicGeometry.HasRingHomProperty.iff_exists_appLE_locally, AlgebraicGeometry.Scheme.Pullback.range_diagonal_subset_diagonalCoverDiagonalRange, AlgebraicGeometry.targetAffineLocally_affineAnd_iff', AlgebraicGeometry.smoothOfRelativeDimension_iff, AlgebraicGeometry.quasiCompact_iff_forall_isAffineOpen, AlgebraicGeometry.HasRingHomProperty.iff_exists_appLE, mem_smoothLocus, AlgebraicGeometry.instSmoothMorphismRestrict, appLE_appIso_inv_apply, appLE_map'_assoc, AlgebraicGeometry.IsAffineOpen.isoSpec_hom_apply, AlgebraicGeometry.Scheme.Opens.ι_app_self, naturality, AlgebraicGeometry.exists_isFinite_morphismRestrict_of_finite_preimage_singleton, AlgebraicGeometry.AffineTargetMorphismProperty.IsLocal.to_basicOpen, AlgebraicGeometry.Scheme.Pullback.Triplet.Spec_ofPointTensor_SpecTensorTo, AlgebraicGeometry.Scheme.Pullback.Triplet.ofPoint_s, AlgebraicGeometry.QuasiCompactCover.isCompactOpenCovered_of_isAffineOpen, AlgebraicGeometry.Scheme.Opens.range_ι, AlgebraicGeometry.IsAffineOpen.range_fromSpec, AlgebraicGeometry.IsOpenImmersion.image_preimage_eq_preimage_image_of_isPullback, AlgebraicGeometry.isClosedImmersion_iff, AlgebraicGeometry.Spec_closedPoint, range_subset_ker_support, finite_preimage, AlgebraicGeometry.Scheme.Opens.ι_apply, AlgebraicGeometry.pullbackRestrictIsoRestrict_inv_fst_assoc, AlgebraicGeometry.Scheme.iso_inv_base_hom_base_apply, continuous, asFiberHom_fiberToSpecResidueField, AlgebraicGeometry.Smooth.exists_isStandardSmooth, AlgebraicGeometry.Scheme.Pullback.Triplet.exists_preimage, isConstructible_image, stalkMap_hom_inv_apply, AlgebraicGeometry.Surjective.surj, AlgebraicGeometry.Scheme.stalkMap_inv_hom, AlgebraicGeometry.Scheme.comp_base_apply, AlgebraicGeometry.Scheme.fromSpecStalk_app, AlgebraicGeometry.isGermInjectiveAt_iff_of_isOpenImmersion, irreducibleComponentsEquiv_apply_coe, AlgebraicGeometry.exists_basicOpen_le_appLE_of_appLE_of_isAffine, AlgebraicGeometry.IsAffineOpen.isoSpec_hom_base_apply, AlgebraicGeometry.IsDominant.denseRange, AlgebraicGeometry.Scheme.homOfLE_base, AlgebraicGeometry.instGeometricallyIntegralMorphismRestrict, AlgebraicGeometry.Flat.iff_flat_stalkMap, AlgebraicGeometry.Scheme.IdealSheafData.subschemeι_app_surjective, AlgebraicGeometry.Scheme.preimage_zeroLocus, AlgebraicGeometry.isIso_iff_stalk_iso, AlgebraicGeometry.Scheme.instIsOverMapStalkMapOverInferInstanceOverClass, AlgebraicGeometry.Spec.map_base, support_ker, AlgebraicGeometry.exists_mem_of_isClosed_of_nonempty, homeomorph_apply, stalkMap_congr, isIso_toPshHom, AlgebraicGeometry.affineOpensRestrict_symm_apply_coe, AlgebraicGeometry.IsOpenImmersion.opensRange_pullback_snd_of_left, AlgebraicGeometry.IsIntegralHom.hasAffineProperty, preimage_mono, AlgebraicGeometry.Scheme.germ_stalkClosedPointTo_Spec, AlgebraicGeometry.exists_map_eq_top, AlgebraicGeometry.IsAffineOpen.isLocalization_stalk', surjective, appLE_comp_appLE_assoc, AlgebraicGeometry.Scheme.IsLocallyDirected.exists_of_pullback_V_V, AlgebraicGeometry.Scheme.SpecMap_presheaf_map_eqToHom, AlgebraicGeometry.pullbackRestrictIsoRestrict_hom_morphismRestrict, AlgebraicGeometry.isFinite_iff, stalkMap_congr_assoc, AlgebraicGeometry.IsOpenImmersion.opensRange_pullback_fst_of_right, AlgebraicGeometry.IsZariskiLocalAtSource.iff_exists_resLE, AlgebraicGeometry.Scheme.stalkMap_inv_hom_assoc, AlgebraicGeometry.Scheme.Cover.RelativeGluingData.preimage_toBase_eq_range_ι, AlgebraicGeometry.locallyQuasiFinite_iff_finite_preimage_singleton, AlgebraicGeometry.exists_app_map_eq_zero_of_isLimit, AlgebraicGeometry.IsOpenImmersion.range_pullbackFst, AlgebraicGeometry.IsAffineOpen.fromSpec_image_zeroLocus, AlgebraicGeometry.Scheme.appLE_comp_appLE, AlgebraicGeometry.Scheme.Cover.instEpiTopCatBaseCommRingCatFromGlued, AlgebraicGeometry.Scheme.affineBasisCover_map_range, resLE_comp_resLE, resLE_appLE, AlgebraicGeometry.Flat.isQuotientMap_of_surjective, AlgebraicGeometry.Scheme.Opens.ι_app, QuasiFiniteAt.isClopen_singleton_asFiber, AlgebraicGeometry.Scheme.Modules.pushforwardCongr_inv_app_app, AlgebraicGeometry.instIsPreimmersionAsFiberHom, isCompact_preimage, AlgebraicGeometry.quasiCompact_iff_spectral, AlgebraicGeometry.Scheme.iso_hom_base_inv_base_apply, AlgebraicGeometry.instUniversallyClosedMorphismRestrict, AlgebraicGeometry.quasiCompact_iff_isSpectralMap, instIsIsoCommRingCatApp, inv_image, AlgebraicGeometry.IsPreimmersion.instMorphismRestrict, comp_base, AlgebraicGeometry.exists_finite_imageι_comp_morphismRestrict_of_finite_image_preimage, AlgebraicGeometry.IsAffineOpen.isCompactOpenCovered, AlgebraicGeometry.IsOpenImmersion.map_ΓIso_hom, map_appLE, AlgebraicGeometry.Scheme.Γevaluation_naturality, AlgebraicGeometry.IsSeparated.instMorphismRestrict, appIso_inv_app_apply, AlgebraicGeometry.HasRingHomProperty.stalkMap_of_respectsIso, AlgebraicGeometry.ExistsHomHomCompEqCompAux.range_g_subset, AlgebraicGeometry.quasiCompact_iff_forall_affine, AlgebraicGeometry.instLocallyOfFinitePresentationMorphismRestrict, AlgebraicGeometry.Scheme.isLocalHom_stalkClosedPointTo, AlgebraicGeometry.IsAffineOpen.fromSpec_preimage_basicOpen', AlgebraicGeometry.IsAffineOpen.app_basicOpen_eq_away_map, AlgebraicGeometry.Scheme.preimage_basicOpen, AlgebraicGeometry.Scheme.Opens.instIsIsoCommRingCatAppLEιTopToScheme, AlgebraicGeometry.IsAffineHom.isAffine_preimage, AlgebraicGeometry.IsAffineOpen.ΓSpecIso_hom_fromSpec_app, SpecMap_residueFieldMap_fromSpecResidueField, AlgebraicGeometry.Scheme.comp_app_assoc, AlgebraicGeometry.Scheme.Pullback.Triplet.ofPoint_y, AlgebraicGeometry.morphismRestrict_app, appLE_comp_appLE, toImage_app, AlgebraicGeometry.Scheme.range_fromSpecResidueField, AlgebraicGeometry.Scheme.SpecMap_stalkMap_fromSpecStalk, stalkMap_inv_hom_apply, AlgebraicGeometry.quasiCompactCover_iff, AlgebraicGeometry.Scheme.image_zeroLocus, appLE_map, fiberToSpecResidueField_apply, AlgebraicGeometry.Scheme.inv_base_hom_base_assoc, stalkMap_hom_inv_assoc, AlgebraicGeometry.Scheme.Pullback.Triplet.ofPoint_SpecTensorTo, AlgebraicGeometry.Scheme.OpenCover.restrict_X, AlgebraicGeometry.Scheme.JointlySurjective.exists_eq, AlgebraicGeometry.IsAffineOpen.fromSpec_app_of_le, AlgebraicGeometry.Scheme.toSpecΓ_preimage_basicOpen, isIrreducible_preimage, app_appIso_inv, AlgebraicGeometry.Scheme.coe_homeoOfIso, normalizationObjIso_hom_val, appLE_eq_app, AlgebraicGeometry.IsOpenImmersion.app_eq_invApp_app_of_comp_eq_aux, AlgebraicGeometry.isClosedMap_iff_specializingMap, AlgebraicGeometry.Scheme.image_preimage_eq_of_isPullback, instIsLocalHomCarrierStalkCommRingCatPresheafCoeContinuousMapCarrierCarrierHomTopCatBaseRingHomHomStalkMap, AlgebraicGeometry.Scheme.GlueData.ι_jointly_surjective, AlgebraicGeometry.Scheme.AffineZariskiSite.PreservesLocalization.colimitDesc_preimage, AlgebraicGeometry.Scheme.stalkClosedPointTo_comp, AlgebraicGeometry.Scheme.Spec_map_residue_apply, AlgebraicGeometry.sourceAffineLocally_morphismRestrict, AlgebraicGeometry.Scheme.residueFieldMap_comp, AlgebraicGeometry.Etale.instMorphismRestrict, AlgebraicGeometry.Scheme.isoSpec_image_zeroLocus, fiberι_fiberHomeo_symm, AlgebraicGeometry.Scheme.comp_coeBase, AlgebraicGeometry.Scheme.IdealSheafData.ideal_map_of_isAffineHom, AlgebraicGeometry.Scheme.Spec_stalkClosedPointTo_fromSpecStalk, le_resLE_preimage_iff, AlgebraicGeometry.Scheme.Modules.pushforward_map_app, AlgebraicGeometry.Scheme.mem_toGrothendieck_smallPretopology, AlgebraicGeometry.Scheme.IsJointlySurjectivePreserving.exists_preimage_fst_triplet_of_prop, stalkMap_inv_hom, inv_app, isQuasiSeparated_preimage, app_injective, Spec_map_residueFieldMap_fromSpecResidueField, AlgebraicGeometry.Scheme.Opens.toSpecΓ_preimage_zeroLocus, AlgebraicGeometry.Scheme.Cover.fromGlued_open_map, AlgebraicGeometry.Scheme.exists_isOpenCover_and_isAffine, AlgebraicGeometry.coprodMk_inr, isProperMap, AlgebraicGeometry.QuasiCompactCover.exists_isAffineOpen_of_isCompact, AlgebraicGeometry.Scheme.stalkMap_germ, asFiberHom_fiberToSpecResidueField_assoc, AlgebraicGeometry.Scheme.Pullback.exists_preimage_pullback, AlgebraicGeometry.Proj.fromOfGlobalSections_preimage_basicOpen, AlgebraicGeometry.IsOpenImmersion.comp_lift, isOpen_smoothLocus, AlgebraicGeometry.Scheme.comp_appLE_assoc, fromNormalization_app, AlgebraicGeometry.Scheme.stalkSpecializes_stalkMap_assoc, AlgebraicGeometry.Scheme.Opens.ι_base_apply, eqToHom_app, isDiscrete_preimage, AlgebraicGeometry.Scheme.homOfLE_apply, AlgebraicGeometry.IsAffineOpen.fromSpec_primeIdealOf, AlgebraicGeometry.Scheme.exists_preimage_of_isPullback, AlgebraicGeometry.Scheme.fromSpecResidueField_apply, AlgebraicGeometry.IsOpenImmersion.lift_app, coe_opensRange, AlgebraicGeometry.Scheme.stalkMap_congr, AlgebraicGeometry.isClosedImmersion_diagonal_restrict_diagonalCoverDiagonalRange, stalkMap_congr_hom, AlgebraicGeometry.Scheme.inv_base_hom_base, AlgebraicGeometry.Scheme.IdealSheafData.subschemeι_apply, AlgebraicGeometry.IsOpenImmersion.affineOpensEquiv_symm_apply_coe, AlgebraicGeometry.Scheme.Spec_stalkClosedPointTo_fromSpecStalk_assoc, AlgebraicGeometry.IsOpenImmersion.map_ΓIso_inv_apply, AlgebraicGeometry.IsAffineOpen.preimage, AlgebraicGeometry.Scheme.Opens.isoOfLE_inv_ι_assoc, AlgebraicGeometry.Spec.map_app, AlgebraicGeometry.Scheme.Cover.fromGlued_injective, AlgebraicGeometry.exists_etale_isCompl_of_quasiFiniteAt, AlgebraicGeometry.Spec.map_base_apply, AlgebraicGeometry.morphismRestrict_id, stalkMap_surjective, AlgebraicGeometry.Scheme.toSpecΓ_base, AlgebraicGeometry.Smooth.iff_forall_exists_isStandardSmooth

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
appLE_comp_appLE 📖	mathematical	AlgebraicGeometry.Scheme.Opens Preorder.toLE PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier CategoryTheory.Functor.obj Preorder.smallCategory TopologicalSpace.Opens TopCat.str TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom'	CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op AlgebraicGeometry.Scheme.Opens appLE AlgebraicGeometry.Scheme AlgebraicGeometry.Scheme.instCategory LE.le.trans AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' Quiver.Hom.le CategoryTheory.Functor.map CategoryTheory.homOfLE	—	LE.le.trans CategoryTheory.Category.assoc naturality_assoc CategoryTheory.Functor.map_comp
appLE_comp_appLE_assoc 📖	mathematical	AlgebraicGeometry.Scheme.Opens Preorder.toLE PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier CategoryTheory.Functor.obj Preorder.smallCategory TopologicalSpace.Opens TopCat.str TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom'	CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op AlgebraicGeometry.Scheme.Opens appLE AlgebraicGeometry.Scheme AlgebraicGeometry.Scheme.instCategory LE.le.trans AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' Quiver.Hom.le CategoryTheory.Functor.map CategoryTheory.homOfLE	—	LE.le.trans CategoryTheory.Category.assoc Mathlib.Tactic.Reassoc.eq_whisker' appLE_comp_appLE
appLE_congr 📖	mathematical	AlgebraicGeometry.Scheme.Opens Preorder.toLE PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier CategoryTheory.Functor.obj Preorder.smallCategory TopologicalSpace.Opens TopCat.str TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom'	CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op AlgebraicGeometry.Scheme.Opens appLE Preorder.toLE AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom'	—	—
appLE_eq_app 📖	mathematical	—	appLE CategoryTheory.Functor.obj AlgebraicGeometry.Scheme.Opens Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' le_rfl app	—	le_rfl app_eq_appLE
appLE_map 📖	mathematical	AlgebraicGeometry.Scheme.Opens Preorder.toLE PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier CategoryTheory.Functor.obj Preorder.smallCategory TopologicalSpace.Opens TopCat.str TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom'	CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op AlgebraicGeometry.Scheme.Opens appLE CategoryTheory.Functor.map LE.le.trans AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier Opposite.unop TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' Quiver.Hom.le Quiver.Hom.unop CategoryTheory.CategoryStruct.toQuiver	—	LE.le.trans appLE.eq_1 CategoryTheory.Category.assoc CategoryTheory.Functor.map_comp
appLE_map' 📖	mathematical	AlgebraicGeometry.Scheme.Opens Preorder.toLE PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier CategoryTheory.Functor.obj Preorder.smallCategory TopologicalSpace.Opens TopCat.str TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom'	CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op AlgebraicGeometry.Scheme.Opens appLE Preorder.toLE AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' CategoryTheory.Functor.map Quiver.Hom.op CategoryTheory.CategoryStruct.toQuiver CategoryTheory.eqToHom	—	appLE_map
appLE_map'_assoc 📖	mathematical	AlgebraicGeometry.Scheme.Opens Preorder.toLE PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier CategoryTheory.Functor.obj Preorder.smallCategory TopologicalSpace.Opens TopCat.str TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom'	CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op AlgebraicGeometry.Scheme.Opens appLE Preorder.toLE AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' CategoryTheory.Functor.map Quiver.Hom.op CategoryTheory.CategoryStruct.toQuiver CategoryTheory.eqToHom	—	CategoryTheory.Category.assoc Mathlib.Tactic.Reassoc.eq_whisker' appLE_map'
appLE_map_assoc 📖	mathematical	AlgebraicGeometry.Scheme.Opens Preorder.toLE PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier CategoryTheory.Functor.obj Preorder.smallCategory TopologicalSpace.Opens TopCat.str TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom'	CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op AlgebraicGeometry.Scheme.Opens appLE CategoryTheory.Functor.map LE.le.trans AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier Opposite.unop TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' Quiver.Hom.le Quiver.Hom.unop CategoryTheory.CategoryStruct.toQuiver	—	LE.le.trans CategoryTheory.Category.assoc Mathlib.Tactic.Reassoc.eq_whisker' appLE_map
app_eq 📖	mathematical	—	app CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op AlgebraicGeometry.Scheme.Opens AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' CategoryTheory.Functor.map Quiver.Hom.op CategoryTheory.CategoryStruct.toQuiver CategoryTheory.eqToHom	—	CategoryTheory.Functor.map_id CategoryTheory.Category.comp_id CategoryTheory.Category.id_comp
app_eq_appLE 📖	mathematical	—	app appLE CategoryTheory.Functor.obj AlgebraicGeometry.Scheme.Opens Preorder.smallCategory TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace TopCat.str PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' le_rfl AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier	—	le_rfl CategoryTheory.Functor.map_id CategoryTheory.Category.comp_id
coe_preimage 📖	mathematical	—	SetLike.coe AlgebraicGeometry.Scheme.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace TopologicalSpace.Opens.instSetLike AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier CategoryTheory.Functor.obj Preorder.smallCategory TopologicalSpace.Opens TopCat.str PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' Set.preimage DFunLike.coe CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier	—	—
comp_app 📖	mathematical	—	app CategoryTheory.CategoryStruct.comp AlgebraicGeometry.Scheme CategoryTheory.Category.toCategoryStruct AlgebraicGeometry.Scheme.instCategory CommRingCat CommRingCat.instCategory CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op AlgebraicGeometry.Scheme.Opens AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom'	—	—
comp_appLE 📖	mathematical	AlgebraicGeometry.Scheme.Opens Preorder.toLE PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier CategoryTheory.Functor.obj Preorder.smallCategory TopologicalSpace.Opens TopCat.str TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' CategoryTheory.CategoryStruct.comp AlgebraicGeometry.Scheme CategoryTheory.Category.toCategoryStruct AlgebraicGeometry.Scheme.instCategory	appLE CategoryTheory.CategoryStruct.comp AlgebraicGeometry.Scheme CategoryTheory.Category.toCategoryStruct AlgebraicGeometry.Scheme.instCategory CommRingCat CommRingCat.instCategory CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op AlgebraicGeometry.Scheme.Opens AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' app	—	le_rfl app_eq_appLE LE.le.trans appLE_comp_appLE
comp_appLE_assoc 📖	mathematical	AlgebraicGeometry.Scheme.Opens Preorder.toLE PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier CategoryTheory.Functor.obj Preorder.smallCategory TopologicalSpace.Opens TopCat.str TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' CategoryTheory.CategoryStruct.comp AlgebraicGeometry.Scheme CategoryTheory.Category.toCategoryStruct AlgebraicGeometry.Scheme.instCategory	CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op AlgebraicGeometry.Scheme.Opens appLE AlgebraicGeometry.Scheme AlgebraicGeometry.Scheme.instCategory AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' app	—	CategoryTheory.Category.assoc Mathlib.Tactic.Reassoc.eq_whisker' comp_appLE
comp_appTop 📖	mathematical	—	appTop CategoryTheory.CategoryStruct.comp AlgebraicGeometry.Scheme CategoryTheory.Category.toCategoryStruct AlgebraicGeometry.Scheme.instCategory CommRingCat CommRingCat.instCategory CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op AlgebraicGeometry.Scheme.Opens Top.top OrderTop.toTop Preorder.toLE SemilatticeSup.toPartialOrder Lattice.toSemilatticeSup CompleteLattice.toLattice TopologicalSpace.Opens.instCompleteLattice AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier BoundedOrder.toOrderTop CompleteLattice.toBoundedOrder	—	—
comp_appTop_assoc 📖	mathematical	—	CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op AlgebraicGeometry.Scheme.Opens Top.top OrderTop.toTop Preorder.toLE SemilatticeSup.toPartialOrder Lattice.toSemilatticeSup CompleteLattice.toLattice TopologicalSpace.Opens.instCompleteLattice AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier BoundedOrder.toOrderTop CompleteLattice.toBoundedOrder appTop AlgebraicGeometry.Scheme AlgebraicGeometry.Scheme.instCategory	—	CategoryTheory.Category.assoc Mathlib.Tactic.Reassoc.eq_whisker' comp_appTop
comp_app_assoc 📖	mathematical	—	CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op AlgebraicGeometry.Scheme.Opens AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' AlgebraicGeometry.Scheme AlgebraicGeometry.Scheme.instCategory app	—	Mathlib.Tactic.Reassoc.eq_whisker' comp_app
comp_apply 📖	mathematical	—	DFunLike.coe AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace TopCat.carrier CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap TopCat.str ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' CategoryTheory.CategoryStruct.comp AlgebraicGeometry.Scheme CategoryTheory.Category.toCategoryStruct AlgebraicGeometry.Scheme.instCategory	—	—
comp_base 📖	mathematical	—	AlgebraicGeometry.PresheafedSpace.Hom.base CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' CategoryTheory.CategoryStruct.comp AlgebraicGeometry.Scheme CategoryTheory.Category.toCategoryStruct AlgebraicGeometry.Scheme.instCategory TopCat TopCat.instCategory AlgebraicGeometry.PresheafedSpace.carrier	—	—
comp_base_assoc 📖	mathematical	—	CategoryTheory.CategoryStruct.comp TopCat CategoryTheory.Category.toCategoryStruct TopCat.instCategory AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' AlgebraicGeometry.Scheme AlgebraicGeometry.Scheme.instCategory	—	CategoryTheory.Category.assoc Mathlib.Tactic.Reassoc.eq_whisker' comp_base
comp_preimage 📖	mathematical	—	CategoryTheory.Functor.obj AlgebraicGeometry.Scheme.Opens Preorder.smallCategory TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace TopCat.str PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' CategoryTheory.CategoryStruct.comp AlgebraicGeometry.Scheme CategoryTheory.Category.toCategoryStruct AlgebraicGeometry.Scheme.instCategory AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier	—	—
comp_toLRSHom 📖	mathematical	—	toLRSHom CategoryTheory.CategoryStruct.comp AlgebraicGeometry.Scheme CategoryTheory.Category.toCategoryStruct AlgebraicGeometry.Scheme.instCategory AlgebraicGeometry.LocallyRingedSpace AlgebraicGeometry.LocallyRingedSpace.instCategory AlgebraicGeometry.Scheme.toLocallyRingedSpace	—	—
comp_toLRSHom_assoc 📖	mathematical	—	CategoryTheory.CategoryStruct.comp AlgebraicGeometry.LocallyRingedSpace CategoryTheory.Category.toCategoryStruct AlgebraicGeometry.LocallyRingedSpace.instCategory AlgebraicGeometry.Scheme.toLocallyRingedSpace toLRSHom AlgebraicGeometry.Scheme AlgebraicGeometry.Scheme.instCategory	—	CategoryTheory.Category.assoc Mathlib.Tactic.Reassoc.eq_whisker' comp_toLRSHom
congr_app 📖	mathematical	—	app CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op AlgebraicGeometry.Scheme.Opens AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' CategoryTheory.Functor.map Quiver.Hom.op CategoryTheory.CategoryStruct.toQuiver CategoryTheory.eqToHom	—	CategoryTheory.Functor.map_id CategoryTheory.Category.comp_id
continuous 📖	mathematical	—	Continuous TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier DFunLike.coe CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap TopCat.str ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom'	—	ContinuousMap.continuous_toFun
copyBase_eq 📖	mathematical	DFunLike.coe AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace TopCat.carrier CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap TopCat.str ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom'	copyBase	—	CommRingCat.Colimits.hasColimits_commRingCat TopCat.Presheaf.ext CommRingCat.hom_ext RingHom.ext TopCat.Presheaf.pushforwardEq_hom_app CategoryTheory.Functor.map_id CategoryTheory.Category.comp_id
eqToHom_app 📖	mathematical	—	app CategoryTheory.eqToHom AlgebraicGeometry.Scheme CategoryTheory.Category.toCategoryStruct AlgebraicGeometry.Scheme.instCategory CommRingCat CommRingCat.instCategory CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op AlgebraicGeometry.Scheme.Opens AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom'	—	—
ext 📖	—	AlgebraicGeometry.PresheafedSpace.Hom.base CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' CategoryTheory.CategoryStruct.comp CategoryTheory.Category.toCategoryStruct CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op AlgebraicGeometry.Scheme.Opens AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier TopologicalSpace.Opens.map app CategoryTheory.Functor.map Quiver.Hom.op CategoryTheory.CategoryStruct.toQuiver CategoryTheory.eqToHom Quiver.Hom TopCat TopCat.instCategory	—	—	AlgebraicGeometry.LocallyRingedSpace.Hom.ext' AlgebraicGeometry.PresheafedSpace.Hom.ext TopCat.Presheaf.ext CategoryTheory.Functor.congr_obj CategoryTheory.eqToHom_app CategoryTheory.eqToHom_op
ext' 📖	—	toLRSHom	—	—	—
germ_stalkMap 📖	mathematical	TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.Scheme.Opens SetLike.instMembership TopologicalSpace.Opens.instSetLike AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier DFunLike.coe CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap TopCat.str ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom'	CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op TopCat.Presheaf.stalk CommRingCat.Colimits.hasColimits_commRingCat DFunLike.coe CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' TopCat.Presheaf.germ stalkMap AlgebraicGeometry.Scheme.Opens AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier TopologicalSpace.Opens.map app	—	AlgebraicGeometry.PresheafedSpace.stalkMap_germ CommRingCat.Colimits.hasColimits_commRingCat
germ_stalkMap_apply 📖	mathematical	TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.Scheme.Opens SetLike.instMembership TopologicalSpace.Opens.instSetLike AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier DFunLike.coe CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap TopCat.str ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom'	DFunLike.coe TopCat.Presheaf.stalk CommRingCat CommRingCat.instCategory CommRingCat.Colimits.hasColimits_commRingCat AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.PresheafedSpace.presheaf TopCat.carrier CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap TopCat.str ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' CommRingCat.carrier RingHom Semiring.toNonAssocSemiring CommSemiring.toSemiring CommRing.toCommSemiring CommRingCat.commRing RingHom.instFunLike CommRingCat.instConcreteCategoryRingHomCarrier stalkMap CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder Opposite.op TopCat.Presheaf.germ AlgebraicGeometry.Scheme.Opens AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier TopologicalSpace.Opens.map app	—	AlgebraicGeometry.PresheafedSpace.stalkMap_germ_apply CommRingCat.Colimits.hasColimits_commRingCat
germ_stalkMap_assoc 📖	mathematical	TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.Scheme.Opens SetLike.instMembership TopologicalSpace.Opens.instSetLike AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier DFunLike.coe CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap TopCat.str ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom'	CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op TopCat.Presheaf.stalk CommRingCat.Colimits.hasColimits_commRingCat DFunLike.coe ContinuousMap CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' TopCat.Presheaf.germ stalkMap AlgebraicGeometry.Scheme.Opens AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier TopologicalSpace.Opens.map app	—	CommRingCat.Colimits.hasColimits_commRingCat CategoryTheory.Category.assoc Mathlib.Tactic.Reassoc.eq_whisker' germ_stalkMap
homeomorph_apply 📖	mathematical	—	DFunLike.coe Homeomorph TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier EquivLike.toFunLike Homeomorph.instEquivLike homeomorph CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap TopCat.str ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom'	—	—
iSup_preimage_eq_top 📖	mathematical	iSup AlgebraicGeometry.Scheme.Opens ConditionallyCompletePartialOrderSup.toSupSet ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup ConditionallyCompleteLattice.toConditionallyCompletePartialOrder CompleteLattice.toConditionallyCompleteLattice TopologicalSpace.Opens.instCompleteLattice TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier Top.top OrderTop.toTop Preorder.toLE PartialOrder.toPreorder SemilatticeSup.toPartialOrder Lattice.toSemilatticeSup CompleteLattice.toLattice BoundedOrder.toOrderTop CompleteLattice.toBoundedOrder	iSup AlgebraicGeometry.Scheme.Opens ConditionallyCompletePartialOrderSup.toSupSet ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup ConditionallyCompleteLattice.toConditionallyCompletePartialOrder CompleteLattice.toConditionallyCompleteLattice TopologicalSpace.Opens.instCompleteLattice TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier CategoryTheory.Functor.obj Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' Top.top OrderTop.toTop Preorder.toLE SemilatticeSup.toPartialOrder Lattice.toSemilatticeSup CompleteLattice.toLattice BoundedOrder.toOrderTop CompleteLattice.toBoundedOrder	—	preimage_iSup
id_app 📖	mathematical	—	app CategoryTheory.CategoryStruct.id AlgebraicGeometry.Scheme CategoryTheory.Category.toCategoryStruct AlgebraicGeometry.Scheme.instCategory CommRingCat CommRingCat.instCategory CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op AlgebraicGeometry.Scheme.Opens	—	—
id_appTop 📖	mathematical	—	appTop CategoryTheory.CategoryStruct.id AlgebraicGeometry.Scheme CategoryTheory.Category.toCategoryStruct AlgebraicGeometry.Scheme.instCategory CommRingCat CommRingCat.instCategory CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op AlgebraicGeometry.Scheme.Opens Top.top OrderTop.toTop Preorder.toLE SemilatticeSup.toPartialOrder Lattice.toSemilatticeSup CompleteLattice.toLattice TopologicalSpace.Opens.instCompleteLattice AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier BoundedOrder.toOrderTop CompleteLattice.toBoundedOrder	—	—
id_base 📖	mathematical	—	AlgebraicGeometry.PresheafedSpace.Hom.base CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' CategoryTheory.CategoryStruct.id AlgebraicGeometry.Scheme CategoryTheory.Category.toCategoryStruct AlgebraicGeometry.Scheme.instCategory TopCat TopCat.instCategory AlgebraicGeometry.PresheafedSpace.carrier	—	—
id_preimage 📖	mathematical	—	CategoryTheory.Functor.obj AlgebraicGeometry.Scheme.Opens Preorder.smallCategory TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace TopCat.str PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' CategoryTheory.CategoryStruct.id AlgebraicGeometry.Scheme CategoryTheory.Category.toCategoryStruct AlgebraicGeometry.Scheme.instCategory	—	—
instIsIsoCommRingCatApp 📖	mathematical	—	CategoryTheory.IsIso CommRingCat CommRingCat.instCategory CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op AlgebraicGeometry.Scheme.Opens AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' app	—	CategoryTheory.NatIso.isIso_app_of_isIso AlgebraicGeometry.PresheafedSpace.c_isIso_of_iso isIso_toPshHom
instIsLocalHomCarrierStalkCommRingCatPresheafCoeContinuousMapCarrierCarrierHomTopCatBaseRingHomHomStalkMap 📖	mathematical	—	IsLocalHom CommRingCat.carrier TopCat.Presheaf.stalk CommRingCat CommRingCat.instCategory CommRingCat.Colimits.hasColimits_commRingCat AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.PresheafedSpace.presheaf DFunLike.coe TopCat.carrier CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap TopCat.str ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' RingHom Semiring.toNonAssocSemiring CommSemiring.toSemiring CommRing.toCommSemiring CommRingCat.commRing MonoidWithZero.toMonoid Semiring.toMonoidWithZero RingHom.instFunLike CommRingCat.Hom.hom stalkMap	—	AlgebraicGeometry.LocallyRingedSpace.Hom.prop
inv_app 📖	mathematical	—	app CategoryTheory.inv AlgebraicGeometry.Scheme AlgebraicGeometry.Scheme.instCategory CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op AlgebraicGeometry.Scheme.Opens TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier CategoryTheory.Functor.map Quiver.Hom.op CategoryTheory.CategoryStruct.toQuiver CategoryTheory.eqToHom instIsIsoCommRingCatApp	—	instIsIsoCommRingCatApp CategoryTheory.IsIso.eq_comp_inv comp_app CategoryTheory.IsIso.hom_inv_id congr_app id_app CategoryTheory.Category.id_comp
inv_appTop 📖	mathematical	—	appTop CategoryTheory.inv AlgebraicGeometry.Scheme AlgebraicGeometry.Scheme.instCategory CommRingCat CommRingCat.instCategory CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op AlgebraicGeometry.Scheme.Opens Top.top OrderTop.toTop Preorder.toLE SemilatticeSup.toPartialOrder Lattice.toSemilatticeSup CompleteLattice.toLattice TopologicalSpace.Opens.instCompleteLattice AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier BoundedOrder.toOrderTop CompleteLattice.toBoundedOrder instIsIsoCommRingCatApp	—	instIsIsoCommRingCatApp inv_app CategoryTheory.Functor.map_id CategoryTheory.Category.id_comp
isIso_base 📖	mathematical	—	CategoryTheory.IsIso TopCat TopCat.instCategory AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom'	—	CategoryTheory.Functor.map_isIso
isIso_toLRSHom 📖	mathematical	—	CategoryTheory.IsIso AlgebraicGeometry.LocallyRingedSpace AlgebraicGeometry.LocallyRingedSpace.instCategory AlgebraicGeometry.Scheme.toLocallyRingedSpace toLRSHom	—	CategoryTheory.Functor.map_isIso
isIso_toPshHom 📖	mathematical	—	CategoryTheory.IsIso AlgebraicGeometry.PresheafedSpace CommRingCat CommRingCat.instCategory AlgebraicGeometry.PresheafedSpace.categoryOfPresheafedSpaces AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.PresheafedSpace.Hom.toPshHom AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom'	—	—
map_appLE 📖	mathematical	AlgebraicGeometry.Scheme.Opens Preorder.toLE PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier CategoryTheory.Functor.obj Preorder.smallCategory TopologicalSpace.Opens TopCat.str TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom'	CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op AlgebraicGeometry.Scheme.Opens CategoryTheory.Functor.map appLE LE.le.trans AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' Quiver.Hom.le Opposite.unop Quiver.Hom.unop CategoryTheory.CategoryStruct.toQuiver	—	LE.le.trans appLE.eq_1 naturality_assoc CategoryTheory.Functor.map_comp
map_appLE' 📖	mathematical	AlgebraicGeometry.Scheme.Opens Preorder.toLE PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier CategoryTheory.Functor.obj Preorder.smallCategory TopologicalSpace.Opens TopCat.str TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom'	CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op AlgebraicGeometry.Scheme.Opens CategoryTheory.Functor.map Quiver.Hom.op CategoryTheory.CategoryStruct.toQuiver AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier CategoryTheory.eqToHom appLE Preorder.toLE TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom'	—	map_appLE
map_appLE'_assoc 📖	mathematical	AlgebraicGeometry.Scheme.Opens Preorder.toLE PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier CategoryTheory.Functor.obj Preorder.smallCategory TopologicalSpace.Opens TopCat.str TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom'	CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op AlgebraicGeometry.Scheme.Opens CategoryTheory.Functor.map Quiver.Hom.op CategoryTheory.CategoryStruct.toQuiver AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier CategoryTheory.eqToHom appLE Preorder.toLE TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom'	—	CategoryTheory.Category.assoc Mathlib.Tactic.Reassoc.eq_whisker' map_appLE'
map_appLE_assoc 📖	mathematical	AlgebraicGeometry.Scheme.Opens Preorder.toLE PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier CategoryTheory.Functor.obj Preorder.smallCategory TopologicalSpace.Opens TopCat.str TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom'	CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op AlgebraicGeometry.Scheme.Opens CategoryTheory.Functor.map appLE LE.le.trans AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' Quiver.Hom.le Opposite.unop Quiver.Hom.unop CategoryTheory.CategoryStruct.toQuiver	—	LE.le.trans CategoryTheory.Category.assoc Mathlib.Tactic.Reassoc.eq_whisker' map_appLE
mem_preimage 📖	mathematical	—	TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.Scheme.Opens SetLike.instMembership TopologicalSpace.Opens.instSetLike AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier CategoryTheory.Functor.obj Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' DFunLike.coe CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap TopCat.str ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier	—	—
naturality 📖	mathematical	—	CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op AlgebraicGeometry.Scheme.Opens AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' CategoryTheory.Functor.map app Opposite.unop Quiver.Hom.op CategoryTheory.CategoryStruct.toQuiver Quiver.Hom.unop	—	CategoryTheory.NatTrans.naturality
naturality_assoc 📖	mathematical	—	CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op AlgebraicGeometry.Scheme.Opens CategoryTheory.Functor.map AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' app Quiver.Hom.op CategoryTheory.CategoryStruct.toQuiver Quiver.Hom.unop	—	CategoryTheory.Category.assoc Mathlib.Tactic.Reassoc.eq_whisker' naturality
preimage_basicOpen 📖	mathematical	—	CategoryTheory.Functor.obj AlgebraicGeometry.Scheme.Opens Preorder.smallCategory TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace TopCat.str PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' AlgebraicGeometry.Scheme.basicOpen AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier DFunLike.coe Opposite CategoryTheory.Category.opposite AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op CommRingCat.carrier CategoryTheory.ConcreteCategory.hom RingHom Semiring.toNonAssocSemiring CommSemiring.toSemiring CommRing.toCommSemiring CommRingCat.commRing RingHom.instFunLike CommRingCat.instConcreteCategoryRingHomCarrier app	—	AlgebraicGeometry.Scheme.preimage_basicOpen
preimage_basicOpen_top 📖	mathematical	—	CategoryTheory.Functor.obj AlgebraicGeometry.Scheme.Opens Preorder.smallCategory TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace TopCat.str PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' AlgebraicGeometry.Scheme.basicOpen Top.top OrderTop.toTop Preorder.toLE SemilatticeSup.toPartialOrder Lattice.toSemilatticeSup CompleteLattice.toLattice TopologicalSpace.Opens.instCompleteLattice AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier BoundedOrder.toOrderTop CompleteLattice.toBoundedOrder DFunLike.coe Opposite CategoryTheory.Category.opposite AlgebraicGeometry.PresheafedSpace.presheaf Opposite.op CommRingCat.carrier CategoryTheory.ConcreteCategory.hom RingHom Semiring.toNonAssocSemiring CommSemiring.toSemiring CommRing.toCommSemiring CommRingCat.commRing RingHom.instFunLike CommRingCat.instConcreteCategoryRingHomCarrier appTop	—	AlgebraicGeometry.Scheme.preimage_basicOpen_top
preimage_bot 📖	mathematical	—	CategoryTheory.Functor.obj AlgebraicGeometry.Scheme.Opens Preorder.smallCategory TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace TopCat.str PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' Bot.bot OrderBot.toBot Preorder.toLE SemilatticeSup.toPartialOrder Lattice.toSemilatticeSup GeneralizedHeytingAlgebra.toLattice HeytingAlgebra.toGeneralizedHeytingAlgebra Order.Frame.toHeytingAlgebra TopologicalSpace.Opens.instFrame AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier HeytingAlgebra.toOrderBot	—	—
preimage_iSup 📖	mathematical	—	CategoryTheory.Functor.obj AlgebraicGeometry.Scheme.Opens Preorder.smallCategory TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace TopCat.str PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' iSup ConditionallyCompletePartialOrderSup.toSupSet ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup ConditionallyCompleteLattice.toConditionallyCompletePartialOrder CompleteLattice.toConditionallyCompleteLattice TopologicalSpace.Opens.instCompleteLattice AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier	—	TopologicalSpace.Opens.ext TopologicalSpace.Opens.coe_iSup Set.preimage_iUnion isOpen_iUnion TopologicalSpace.Opens.is_open'
preimage_iSup_eq_top 📖	mathematical	iSup AlgebraicGeometry.Scheme.Opens ConditionallyCompletePartialOrderSup.toSupSet ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup ConditionallyCompleteLattice.toConditionallyCompletePartialOrder CompleteLattice.toConditionallyCompleteLattice TopologicalSpace.Opens.instCompleteLattice TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier Top.top OrderTop.toTop Preorder.toLE PartialOrder.toPreorder SemilatticeSup.toPartialOrder Lattice.toSemilatticeSup CompleteLattice.toLattice BoundedOrder.toOrderTop CompleteLattice.toBoundedOrder	iSup AlgebraicGeometry.Scheme.Opens ConditionallyCompletePartialOrderSup.toSupSet ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup ConditionallyCompleteLattice.toConditionallyCompletePartialOrder CompleteLattice.toConditionallyCompleteLattice TopologicalSpace.Opens.instCompleteLattice TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier CategoryTheory.Functor.obj Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' Top.top OrderTop.toTop Preorder.toLE SemilatticeSup.toPartialOrder Lattice.toSemilatticeSup CompleteLattice.toLattice BoundedOrder.toOrderTop CompleteLattice.toBoundedOrder	—	iSup_preimage_eq_top
preimage_inf 📖	mathematical	—	CategoryTheory.Functor.obj AlgebraicGeometry.Scheme.Opens Preorder.smallCategory TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace TopCat.str PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' SemilatticeInf.toMin Lattice.toSemilatticeInf ConditionallyCompleteLattice.toLattice CompleteLattice.toConditionallyCompleteLattice TopologicalSpace.Opens.instCompleteLattice AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier	—	—
preimage_le_preimage_of_le 📖	mathematical	AlgebraicGeometry.Scheme.Opens Preorder.toLE PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier	AlgebraicGeometry.Scheme.Opens Preorder.toLE PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier CategoryTheory.Functor.obj Preorder.smallCategory TopologicalSpace.Opens TopCat.str TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom'	—	preimage_mono
preimage_mono 📖	mathematical	AlgebraicGeometry.Scheme.Opens Preorder.toLE PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier	AlgebraicGeometry.Scheme.Opens Preorder.toLE PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier CategoryTheory.Functor.obj Preorder.smallCategory TopologicalSpace.Opens TopCat.str TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom'	—	—
preimage_sup 📖	mathematical	—	CategoryTheory.Functor.obj AlgebraicGeometry.Scheme.Opens Preorder.smallCategory TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace TopCat.str PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' SemilatticeSup.toMax Lattice.toSemilatticeSup ConditionallyCompleteLattice.toLattice CompleteLattice.toConditionallyCompleteLattice TopologicalSpace.Opens.instCompleteLattice AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier	—	—
preimage_top 📖	mathematical	—	CategoryTheory.Functor.obj AlgebraicGeometry.Scheme.Opens Preorder.smallCategory TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace TopCat.str PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' Top.top OrderTop.toTop Preorder.toLE SemilatticeSup.toPartialOrder Lattice.toSemilatticeSup CompleteLattice.toLattice TopologicalSpace.Opens.instCompleteLattice AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier BoundedOrder.toOrderTop CompleteLattice.toBoundedOrder	—	—
stalkMap_comp 📖	mathematical	—	stalkMap CategoryTheory.CategoryStruct.comp AlgebraicGeometry.Scheme CategoryTheory.Category.toCategoryStruct AlgebraicGeometry.Scheme.instCategory CommRingCat CommRingCat.instCategory TopCat.Presheaf.stalk CommRingCat.Colimits.hasColimits_commRingCat AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.PresheafedSpace.presheaf DFunLike.coe TopCat.carrier CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap TopCat.str ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom'	—	AlgebraicGeometry.PresheafedSpace.stalkMap.comp CommRingCat.Colimits.hasColimits_commRingCat
stalkMap_congr 📖	mathematical	—	CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory TopCat.Presheaf.stalk CommRingCat.Colimits.hasColimits_commRingCat AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.PresheafedSpace.presheaf DFunLike.coe TopCat.carrier CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap TopCat.str ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' stalkMap CategoryTheory.Iso.hom TopCat.Presheaf.stalkCongr Inseparable.of_eq Quiver.Hom AlgebraicGeometry.Scheme CategoryTheory.CategoryStruct.toQuiver AlgebraicGeometry.Scheme.instCategory	—	AlgebraicGeometry.LocallyRingedSpace.stalkMap_congr
stalkMap_congr_assoc 📖	mathematical	—	CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory TopCat.Presheaf.stalk CommRingCat.Colimits.hasColimits_commRingCat AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.PresheafedSpace.presheaf DFunLike.coe ContinuousMap TopCat.carrier TopCat.str CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' stalkMap CategoryTheory.Iso.hom TopCat.Presheaf.stalkCongr Inseparable.of_eq Quiver.Hom AlgebraicGeometry.Scheme CategoryTheory.CategoryStruct.toQuiver AlgebraicGeometry.Scheme.instCategory	—	CommRingCat.Colimits.hasColimits_commRingCat Inseparable.of_eq CategoryTheory.Category.assoc Mathlib.Tactic.Reassoc.eq_whisker' stalkMap_congr
stalkMap_congr_hom 📖	mathematical	—	stalkMap CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory TopCat.Presheaf.stalk CommRingCat.Colimits.hasColimits_commRingCat AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.PresheafedSpace.presheaf DFunLike.coe TopCat.carrier CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap TopCat.str ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' CategoryTheory.Iso.hom TopCat.Presheaf.stalkCongr Inseparable.of_eq Quiver.Hom AlgebraicGeometry.Scheme CategoryTheory.CategoryStruct.toQuiver AlgebraicGeometry.Scheme.instCategory	—	AlgebraicGeometry.LocallyRingedSpace.stalkMap_congr_hom
stalkMap_congr_hom_assoc 📖	mathematical	—	CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory TopCat.Presheaf.stalk CommRingCat.Colimits.hasColimits_commRingCat AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.PresheafedSpace.presheaf DFunLike.coe ContinuousMap TopCat.carrier TopCat.str CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' stalkMap CategoryTheory.Iso.hom TopCat.Presheaf.stalkCongr Inseparable.of_eq Quiver.Hom AlgebraicGeometry.Scheme CategoryTheory.CategoryStruct.toQuiver AlgebraicGeometry.Scheme.instCategory	—	CommRingCat.Colimits.hasColimits_commRingCat Inseparable.of_eq CategoryTheory.Category.assoc Mathlib.Tactic.Reassoc.eq_whisker' stalkMap_congr_hom
stalkMap_congr_point 📖	mathematical	—	CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory TopCat.Presheaf.stalk CommRingCat.Colimits.hasColimits_commRingCat AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.PresheafedSpace.presheaf DFunLike.coe TopCat.carrier CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap TopCat.str ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' stalkMap CategoryTheory.Iso.hom TopCat.Presheaf.stalkCongr Inseparable.of_eq	—	AlgebraicGeometry.LocallyRingedSpace.stalkMap_congr_point
stalkMap_congr_point_assoc 📖	mathematical	—	CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory TopCat.Presheaf.stalk CommRingCat.Colimits.hasColimits_commRingCat AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.PresheafedSpace.presheaf DFunLike.coe ContinuousMap TopCat.carrier TopCat.str CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' stalkMap CategoryTheory.Iso.hom TopCat.Presheaf.stalkCongr Inseparable.of_eq	—	CommRingCat.Colimits.hasColimits_commRingCat Inseparable.of_eq CategoryTheory.Category.assoc Mathlib.Tactic.Reassoc.eq_whisker' stalkMap_congr_point
stalkMap_hom_inv 📖	mathematical	—	CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory TopCat.Presheaf.stalk CommRingCat.Colimits.hasColimits_commRingCat AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.PresheafedSpace.presheaf DFunLike.coe TopCat.carrier CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap TopCat.str ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' CategoryTheory.Iso.hom AlgebraicGeometry.Scheme AlgebraicGeometry.Scheme.instCategory CategoryTheory.Iso.inv stalkMap TopCat.Presheaf.stalkCongr Inseparable.of_eq	—	AlgebraicGeometry.LocallyRingedSpace.stalkMap_hom_inv
stalkMap_hom_inv_apply 📖	mathematical	—	DFunLike.coe TopCat.Presheaf.stalk CommRingCat CommRingCat.instCategory CommRingCat.Colimits.hasColimits_commRingCat AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.PresheafedSpace.presheaf TopCat.carrier CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap TopCat.str ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' CategoryTheory.Iso.inv AlgebraicGeometry.Scheme AlgebraicGeometry.Scheme.instCategory CommRingCat.carrier RingHom Semiring.toNonAssocSemiring CommSemiring.toSemiring CommRing.toCommSemiring CommRingCat.commRing RingHom.instFunLike CommRingCat.instConcreteCategoryRingHomCarrier stalkMap CategoryTheory.Iso.hom TopCat.Presheaf.stalkCongr Inseparable.of_eq	—	CommRingCat.Colimits.hasColimits_commRingCat DFunLike.congr_fun Inseparable.of_eq CommRingCat.hom_ext_iff stalkMap_hom_inv
stalkMap_hom_inv_assoc 📖	mathematical	—	CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory TopCat.Presheaf.stalk CommRingCat.Colimits.hasColimits_commRingCat AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.PresheafedSpace.presheaf DFunLike.coe ContinuousMap TopCat.carrier TopCat.str CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' CategoryTheory.Iso.hom AlgebraicGeometry.Scheme AlgebraicGeometry.Scheme.instCategory CategoryTheory.Iso.inv stalkMap TopCat.Presheaf.stalkCongr Inseparable.of_eq	—	CommRingCat.Colimits.hasColimits_commRingCat Inseparable.of_eq CategoryTheory.Category.assoc Mathlib.Tactic.Reassoc.eq_whisker' stalkMap_hom_inv
stalkMap_id 📖	mathematical	—	stalkMap CategoryTheory.CategoryStruct.id AlgebraicGeometry.Scheme CategoryTheory.Category.toCategoryStruct AlgebraicGeometry.Scheme.instCategory CommRingCat CommRingCat.instCategory TopCat.Presheaf.stalk CommRingCat.Colimits.hasColimits_commRingCat AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.PresheafedSpace.presheaf	—	AlgebraicGeometry.PresheafedSpace.stalkMap.id CommRingCat.Colimits.hasColimits_commRingCat
stalkMap_inv_hom 📖	mathematical	—	CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory TopCat.Presheaf.stalk CommRingCat.Colimits.hasColimits_commRingCat AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.PresheafedSpace.presheaf DFunLike.coe TopCat.carrier CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap TopCat.str ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' CategoryTheory.Iso.inv AlgebraicGeometry.Scheme AlgebraicGeometry.Scheme.instCategory CategoryTheory.Iso.hom stalkMap TopCat.Presheaf.stalkCongr Inseparable.of_eq	—	AlgebraicGeometry.LocallyRingedSpace.stalkMap_inv_hom
stalkMap_inv_hom_apply 📖	mathematical	—	DFunLike.coe TopCat.Presheaf.stalk CommRingCat CommRingCat.instCategory CommRingCat.Colimits.hasColimits_commRingCat AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.PresheafedSpace.presheaf TopCat.carrier CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap TopCat.str ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' CategoryTheory.Iso.hom AlgebraicGeometry.Scheme AlgebraicGeometry.Scheme.instCategory CommRingCat.carrier RingHom Semiring.toNonAssocSemiring CommSemiring.toSemiring CommRing.toCommSemiring CommRingCat.commRing RingHom.instFunLike CommRingCat.instConcreteCategoryRingHomCarrier stalkMap CategoryTheory.Iso.inv TopCat.Presheaf.stalkCongr Inseparable.of_eq	—	CommRingCat.Colimits.hasColimits_commRingCat DFunLike.congr_fun Inseparable.of_eq CommRingCat.hom_ext_iff stalkMap_inv_hom
stalkMap_inv_hom_assoc 📖	mathematical	—	CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory TopCat.Presheaf.stalk CommRingCat.Colimits.hasColimits_commRingCat AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.PresheafedSpace.presheaf DFunLike.coe ContinuousMap TopCat.carrier TopCat.str CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' CategoryTheory.Iso.inv AlgebraicGeometry.Scheme AlgebraicGeometry.Scheme.instCategory CategoryTheory.Iso.hom stalkMap TopCat.Presheaf.stalkCongr Inseparable.of_eq	—	CommRingCat.Colimits.hasColimits_commRingCat Inseparable.of_eq CategoryTheory.Category.assoc Mathlib.Tactic.Reassoc.eq_whisker' stalkMap_inv_hom
stalkSpecializes_stalkMap 📖	mathematical	Specializes TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier	CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory TopCat.Presheaf.stalk CommRingCat.Colimits.hasColimits_commRingCat AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.PresheafedSpace.presheaf DFunLike.coe ContinuousMap TopCat.carrier TopCat.str ContinuousMap.instFunLike TopCat.Hom.hom AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' TopCat.Presheaf.stalkSpecializes ContinuousMap.map_specializes stalkMap CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory TopCat.instConcreteCategoryContinuousMapCarrier	—	AlgebraicGeometry.PresheafedSpace.stalkMap.stalkSpecializes_stalkMap CommRingCat.Colimits.hasColimits_commRingCat
stalkSpecializes_stalkMap_apply 📖	mathematical	Specializes TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier	DFunLike.coe TopCat.Presheaf.stalk CommRingCat CommRingCat.instCategory CommRingCat.Colimits.hasColimits_commRingCat AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.PresheafedSpace.presheaf TopCat.carrier CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap TopCat.str ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' CommRingCat.carrier RingHom Semiring.toNonAssocSemiring CommSemiring.toSemiring CommRing.toCommSemiring CommRingCat.commRing RingHom.instFunLike CommRingCat.instConcreteCategoryRingHomCarrier stalkMap TopCat.Hom.hom TopCat.Presheaf.stalkSpecializes ContinuousMap.map_specializes	—	CommRingCat.Colimits.hasColimits_commRingCat DFunLike.congr_fun ContinuousMap.map_specializes CommRingCat.hom_ext_iff stalkSpecializes_stalkMap
stalkSpecializes_stalkMap_assoc 📖	mathematical	Specializes TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier	CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory TopCat.Presheaf.stalk CommRingCat.Colimits.hasColimits_commRingCat AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.PresheafedSpace.presheaf DFunLike.coe ContinuousMap TopCat.carrier TopCat.str ContinuousMap.instFunLike TopCat.Hom.hom AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom toLRSHom' TopCat.Presheaf.stalkSpecializes ContinuousMap.map_specializes stalkMap	—	CommRingCat.Colimits.hasColimits_commRingCat ContinuousMap.map_specializes CategoryTheory.Category.assoc Mathlib.Tactic.Reassoc.eq_whisker' stalkSpecializes_stalkMap

AlgebraicGeometry.Scheme.Hom.Simps

Definitions

Name	Category	Theorems
toLRSHom 📖	CompOp	—

AlgebraicGeometry.Scheme.id

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
base 📖	mathematical	—	AlgebraicGeometry.PresheafedSpace.Hom.base CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.LocallyRingedSpace.Hom.toHom AlgebraicGeometry.Scheme.Hom.toLRSHom' CategoryTheory.CategoryStruct.id AlgebraicGeometry.Scheme CategoryTheory.Category.toCategoryStruct AlgebraicGeometry.Scheme.instCategory TopCat TopCat.instCategory AlgebraicGeometry.PresheafedSpace.carrier	—	AlgebraicGeometry.Scheme.Hom.id_base

AlgebraicGeometry.Spec

Definitions

Name	Category	Theorems
map 📖	CompOp	229 math math: AlgebraicGeometry.Scheme.toSpecΓ_apply, AlgebraicGeometry.AffineSpace.map_Spec_map, AlgebraicGeometry.IsPreimmersion.of_isLocalization, AlgebraicGeometry.Scheme.ΓSpecIso_inv_naturality_assoc, map_eq_id, AlgebraicGeometry.IsAffineOpen.map_fromSpec_assoc, AlgebraicGeometry.Scheme.Pullback.Triplet.specTensorTo_fst, AlgebraicGeometry.Scheme.Spec_map_stalkMap_fromSpecStalk, AlgebraicGeometry.IsPreimmersion.mk_SpecMap, AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv, AlgebraicGeometry.IsAffineOpen.map_fromSpec, AlgebraicGeometry.Scheme.germ_stalkClosedPointTo_Spec_fromSpecStalk_assoc, AlgebraicGeometry.Scheme.Opens.toSpecΓ_naturality, AlgebraicGeometry.Scheme.Opens.toSpecΓ_SpecMap_presheaf_map_top, AlgebraicGeometry.Proj.fromOfGlobalSections_toSpecZero_assoc, AlgebraicGeometry.AffineSpace.toSpecMvPolyIntEquiv_symm_apply, AlgebraicGeometry.Scheme.fromSpecStalk_toSpecΓ_assoc, AlgebraicGeometry.isIso_SpecMap_iff, map_surjective, AlgebraicGeometry.HasAffineProperty.SpecMap_iff_of_affineAnd, AlgebraicGeometry.diagonal_SpecMap, AlgebraicGeometry.ι_sigmaSpec, map_injective, AlgebraicGeometry.coprodSpec_inr, Algebra.FormallyUnramified.isOpenImmersion_SpecMap_lmul, AlgebraicGeometry.Scheme.Spec_map_stalkSpecializes_fromSpecStalk, AlgebraicGeometry.SpecMap_preimage_basicOpen, AlgebraicGeometry.Scheme.Opens.toSpecΓ_SpecMap_map, AlgebraicGeometry.Scheme.basic_open_isOpenImmersion, AlgebraicGeometry.isPullback_fiberToSpecResidueField_of_isPullback, AlgebraicGeometry.Scheme.Pullback.Triplet.isPullback_SpecMap_tensor, AlgebraicGeometry.Scheme.Pullback.tensorCongr_SpecTensorTo, AlgebraicGeometry.Scheme.Hom.SpecMap_residueFieldMap_fromSpecResidueField_assoc, AlgebraicGeometry.Scheme.Pullback.carrierEquiv_eq_iff, AlgebraicGeometry.Scheme.germ_stalkClosedPointTo_Spec_fromSpecStalk, AlgebraicGeometry.Scheme.isOpenImmersion_SpecMap_localizationAway, AlgebraicGeometry.Scheme.instIsOpenImmersionMapOfHomAwayAlgebraMap, AlgebraicGeometry.Scheme.toSpecΓ_naturality_assoc, AlgebraicGeometry.SpecMap_ΓSpecIso_hom, AlgebraicGeometry.stalkwise_SpecMap_iff, AlgebraicGeometry.Scheme.Pullback.Triplet.specTensorTo_fst_assoc, AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv_over_assoc, AlgebraicGeometry.Scheme.isoSpec_hom_naturality, AlgebraicGeometry.IsPreimmersion.mk_Spec_map, AlgebraicGeometry.Scheme.toSpecΓ_naturality, AlgebraicGeometry.Scheme.Hom.ι_toNormalization, AlgebraicGeometry.Proj.pullbackAwayιIso_hom_SpecMap_awayMap_right_assoc, AlgebraicGeometry.isSheaf_propQCTopology_iff, AlgebraicGeometry.Scheme.Opens.toSpecΓ_naturality_assoc, AlgebraicGeometry.IsAffineOpen.isoSpec_inv, AlgebraicGeometry.Scheme.Hom.liftQuotient_comp, AlgebraicGeometry.Proj.pullbackAwayιIso_inv_snd_assoc, AlgebraicGeometry.Scheme.Opens.toSpecΓ_SpecMap_presheaf_map_assoc, AlgebraicGeometry.Proj.SpecMap_awayMap_awayι_assoc, AlgebraicGeometry.Scheme.isoSpec_hom_naturality_assoc, AlgebraicGeometry.pullbackSpecIso_inv_snd, AlgebraicGeometry.Proj.pullbackAwayιIso_inv_fst, AlgebraicGeometry.Proj.awayι_comp_map_assoc, AlgebraicGeometry.SpecMap_ΓSpecIso_inv_toSpecΓ, AlgebraicGeometry.Proj.basicOpenToSpec_SpecMap_awayMap, AlgebraicGeometry.Scheme.Pullback.Triplet.Spec_map_tensor_isPullback, AlgebraicGeometry.pullbackSpecIso_hom_base, AlgebraicGeometry.IsPreimmersion.Spec_map_iff, AlgebraicGeometry.Scheme.Spec_map_presheaf_map_eqToHom, AlgebraicGeometry.Scheme.fromSpecStalk_toSpecΓ, AlgebraicGeometry.Scheme.Spec_map_stalkMap_fromSpecStalk_assoc, AlgebraicGeometry.Scheme.Opens.toSpecΓ_SpecMap_presheaf_map_top_assoc, map_id, AlgebraicGeometry.Scheme.Pullback.Triplet.specTensorTo_snd, map_appLE, AlgebraicGeometry.Scheme.SpecMap_stalkSpecializes_fromSpecStalk, AlgebraicGeometry.Scheme.descResidueField_fromSpecResidueField, map_preimage_unop, AlgebraicGeometry.ΓSpecIso_obj_hom, AlgebraicGeometry.Proj.pullbackAwayιIso_inv_snd, AlgebraicGeometry.Proj.pullbackAwayιIso_inv_fst_assoc, AlgebraicGeometry.pullbackSpecIso_inv_fst'_assoc, AlgebraicGeometry.isSheaf_type_propQCTopology_iff, AlgebraicGeometry.Scheme.Opens.toSpecΓ_SpecMap_map_assoc, AlgebraicGeometry.Scheme.instIsOverMapResidueFieldMapOverInferInstanceOverClass, AlgebraicGeometry.Scheme.Pullback.Triplet.specTensorTo_snd_assoc, AlgebraicGeometry.Scheme.Pullback.ofPointTensor_SpecTensorTo_assoc, AlgebraicGeometry.Proj.pullbackAwayιIso_hom_SpecMap_awayMap_right, AlgebraicGeometry.specTargetImageFactorization_comp_assoc, AlgebraicGeometry.isOpenImmersion_SpecMap_iff_of_surjective, AlgebraicGeometry.Scheme.Opens.toSpecΓ_SpecMap_presheaf_map, AlgebraicGeometry.IsClosedImmersion.spec_of_quotient_mk, AlgebraicGeometry.Scheme.isoSpec_inv_naturality_assoc, map_comp_assoc, AlgebraicGeometry.Scheme.SpecMap_stalkMap_fromSpecStalk_assoc, AlgebraicGeometry.Scheme.Pullback.tensorCongr_SpecTensorTo_assoc, AlgebraicGeometry.Scheme.isoSpec_Spec_hom, AlgebraicGeometry.Proj.pullbackAwayιIso_hom_SpecMap_awayMap_left, fromSpecStalk_eq', AlgebraicGeometry.pullbackSpecIso_inv_fst, AlgebraicGeometry.IsClosedImmersion.Spec_map_residue, AlgebraicGeometry.Scheme.SpecMap_stalkSpecializes_fromSpecStalk_assoc, AlgebraicGeometry.Scheme.isPullback_toSpecΓ_toSpecΓ, AlgebraicGeometry.Scheme.ΓSpecIso_naturality, AlgebraicGeometry.Proj.awayι_toSpecZero, map_apply, AlgebraicGeometry.isPullback_SpecMap_of_isPushout, AlgebraicGeometry.Scheme.Opens.toSpecΓ_SpecMap_appLE_assoc, AlgebraicGeometry.Proj.awayι_toSpecZero_assoc, AlgebraicGeometry.ValuativeCommSq.commSq, AlgebraicGeometry.Scheme.residueFieldCongr_fromSpecResidueField, AlgebraicGeometry.Scheme.instIsOverMapStalkSpecializesCommRingCatPresheaf, AlgebraicGeometry.Scheme.isoSpec_inv_naturality, AlgebraicGeometry.specTargetImageFactorization_comp, AlgebraicGeometry.Proj.pullbackAwayιIso_hom_SpecMap_awayMap_left_assoc, AlgebraicGeometry.AffineSpace.map_SpecMap, AlgebraicGeometry.Scheme.preimage_opensRange_toSpecΓ, AlgebraicGeometry.Proj.fromOfGlobalSections_toSpecZero, AlgebraicGeometry.AffineSpace.SpecIso_inv_over_assoc, AlgebraicGeometry.IsClosedImmersion.spec_of_surjective, AlgebraicGeometry.toSpecΓ_SpecMap_ΓSpecIso_inv_assoc, AlgebraicGeometry.pullbackSpecIso_inv_snd_assoc, AlgebraicGeometry.isPullback_Spec_map_pushout, AlgebraicGeometry.Scheme.Spec_map_stalkSpecializes_fromSpecStalk_assoc, AlgebraicGeometry.SurjectiveOnStalks.Spec_iff, AlgebraicGeometry.Scheme.descResidueField_stalkClosedPointTo_fromSpecResidueField, AlgebraicGeometry.coprodSpec_inr_assoc, AlgebraicGeometry.Scheme.Spec.map_residueFieldIso_inv_eq_fromSpecResidueField, AlgebraicGeometry.IsClosedImmersion.Spec_iff, AlgebraicGeometry.stalkwise_Spec_map_iff, AlgebraicGeometry.pullbackSpecIso_inv_fst_assoc, AlgebraicGeometry.Scheme.Pullback.ofPointTensor_SpecTensorTo, AlgebraicGeometry.Scheme.Hom.ι_toNormalization_assoc, AlgebraicGeometry.Scheme.Hom.Spec_map_residueFieldMap_fromSpecResidueField_assoc, AlgebraicGeometry.HasRingHomProperty.of_isLocalAtSource_of_isLocalAtTarget, AlgebraicGeometry.IsAffineOpen.primeIdealOf_eq_map_closedPoint, AlgebraicGeometry.instIsOpenImmersionMapOfHomForallEvalRingHom, AlgebraicGeometry.IsFinite.SpecMap_iff, AlgebraicGeometry.Proj.basicOpenToSpec_SpecMap_awayMap_assoc, AlgebraicGeometry.toSpecΓ_SpecMap_ΓSpecIso_inv, AlgebraicGeometry.pullbackSpecIso_hom_fst, AlgebraicGeometry.HasRingHomProperty.isLocal_ringHomProperty_of_isZariskiLocalAtSource_of_isZariskiLocalAtTarget, AlgebraicGeometry.Scheme.ΓSpecIso_inv_naturality, AlgebraicGeometry.IsPreimmersion.SpecMap_iff, map_comp, AlgebraicGeometry.coprodSpec_inl_assoc, AlgebraicGeometry.Scheme.Opens.fromSpecStalkOfMem_toSpecΓ, AlgebraicGeometry.Scheme.isoSpec_Spec_inv, map_eqToHom, AlgebraicGeometry.Proj.SpecMap_awayMap_awayι, AlgebraicGeometry.basicOpenIsoSpecAway_inv_homOfLE_assoc, AlgebraicGeometry.Scheme.Spec_fromSpecStalk', AlgebraicGeometry.Scheme.Spec.map_residueFieldIso_inv_eq_fromSpecResidueField_assoc, AlgebraicGeometry.ι_sigmaSpec_assoc, AlgebraicGeometry.HasRingHomProperty.Spec_iff, AlgebraicGeometry.AffineSpace.isoOfIsAffine_hom, AlgebraicGeometry.Scheme.Hom.ι_fromNormalization_assoc, AlgebraicGeometry.basicOpenIsoSpecAway_inv_homOfLE, AlgebraicGeometry.Scheme.SpecMap_residue_apply, AlgebraicGeometry.coprodSpec_inl, AlgebraicGeometry.Scheme.instIsPreimmersionMapResidue, AlgebraicGeometry.IsAffineOpen.isoSpec_hom_apply, preimage_map, AlgebraicGeometry.Scheme.Pullback.Triplet.Spec_ofPointTensor_SpecTensorTo, AlgebraicGeometry.IsSeparated.instMap, map_inv, AlgebraicGeometry.Spec_closedPoint, AlgebraicGeometry.IsIntegralHom.SpecMap_iff, AlgebraicGeometry.pullbackSpecIso_hom_base_assoc, AlgebraicGeometry.isPullback_SpecMap_pushout, AlgebraicGeometry.Scheme.Opens.fromSpecStalkOfMem_toSpecΓ_assoc, AlgebraicGeometry.Scheme.Hom.asFiberHom_fiberToSpecResidueField, AlgebraicGeometry.Scheme.affineOpenCoverOfSpanRangeEqTop_f, AlgebraicGeometry.SpecMap_residueFieldIsoBase_inv, AlgebraicGeometry.SpecMap_residueFieldIsoBase_inv_assoc, AlgebraicGeometry.IsAffineOpen.isoSpec_hom_base_apply, AlgebraicGeometry.pullbackSpecIso_hom_fst'_assoc, AlgebraicGeometry.Scheme.instIsOverMapStalkMapOverInferInstanceOverClass, AlgebraicGeometry.geometrically_iff_of_commRing_of_isClosedUnderIsomorphisms, AlgebraicGeometry.IsAffineOpen.SpecMap_appLE_fromSpec_assoc, map_base, AlgebraicGeometry.Scheme.instIsOverMapHomCommRingCatResidueFieldCongr, AlgebraicGeometry.SpecToEquivOfLocalRing_symm_apply, AlgebraicGeometry.pullbackSpecIso_inv_fst', AlgebraicGeometry.IsAffineOpen.Spec_map_appLE_fromSpec, AlgebraicGeometry.Scheme.germ_stalkClosedPointTo_Spec, AlgebraicGeometry.Scheme.SpecMap_presheaf_map_eqToHom, AlgebraicGeometry.flat_and_surjective_SpecMap_iff, AlgebraicGeometry.IsAffineOpen.Spec_map_appLE_fromSpec_assoc, AlgebraicGeometry.diagonal_Spec_map, map_preimage, AlgebraicGeometry.IsClosedImmersion.SpecMap_residue, AlgebraicGeometry.Scheme.ΓSpecIso_naturality_assoc, AlgebraicGeometry.IsAffineOpen.fromSpec_toSpecΓ_assoc, AlgebraicGeometry.AffineSpace.SpecIso_inv_over, AlgebraicGeometry.Scheme.Spec_map, AlgebraicGeometry.Scheme.residueFieldCongr_fromSpecResidueField_assoc, AlgebraicGeometry.Scheme.Hom.ι_fromNormalization, AlgebraicGeometry.IsOpenImmersion.of_isLocalization, fromSpecStalk_eq, AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv_over, AlgebraicGeometry.pullbackSpecIso_hom_fst', AlgebraicGeometry.Scheme.IdealSheafData.glueDataObjι_ι, AlgebraicGeometry.isPullback_Spec_map_isPushout, AlgebraicGeometry.Scheme.Hom.SpecMap_residueFieldMap_fromSpecResidueField, AlgebraicGeometry.IsAffineOpen.SpecMap_appLE_fromSpec, AlgebraicGeometry.Scheme.SpecMap_stalkMap_fromSpecStalk, AlgebraicGeometry.instIsIsoSchemeMapOfCommRingCat, AlgebraicGeometry.spread_out_of_isGermInjective, AlgebraicGeometry.pullbackSpecIso_hom_snd, map_inj, AlgebraicGeometry.Scheme.Opens.toSpecΓ_SpecMap_appLE, AlgebraicGeometry.IsAffineOpen.fromSpec_toSpecΓ, AlgebraicGeometry.Scheme.Spec_map_residue_apply, AlgebraicGeometry.Scheme.Spec_stalkClosedPointTo_fromSpecStalk, AlgebraicGeometry.Scheme.localRingHom_comp_stalkIso, AlgebraicGeometry.Scheme.Hom.Spec_map_residueFieldMap_fromSpecResidueField, AlgebraicGeometry.Scheme.Pullback.Triplet.Spec_map_tensorInl_fromSpecResidueField, AlgebraicGeometry.Scheme.Hom.asFiberHom_fiberToSpecResidueField_assoc, AlgebraicGeometry.Scheme.Hom.liftQuotient_comp_assoc, AlgebraicGeometry.Scheme.localRingHom_comp_stalkIso_apply, AlgebraicGeometry.pullbackSpecIso_hom_fst_assoc, AlgebraicGeometry.Spec_stalkClosedPointIso, AlgebraicGeometry.Scheme.Pullback.Triplet.SpecMap_tensorInl_fromSpecResidueField, AlgebraicGeometry.Proj.awayι_comp_map, AlgebraicGeometry.HasRingHomProperty.of_isZariskiLocalAtSource_of_isZariskiLocalAtTarget, homEquiv_symm_apply, AlgebraicGeometry.Scheme.Spec_stalkClosedPointTo_fromSpecStalk_assoc, AlgebraicGeometry.SpecMap_ΓSpecIso_inv_toSpecΓ_assoc, map_app, AlgebraicGeometry.Scheme.Spec_fromSpecStalk, map_base_apply, AlgebraicGeometry.Scheme.toSpecΓ_base, AlgebraicGeometry.pullbackSpecIso_hom_snd_assoc

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
map_app 📖	mathematical	—	AlgebraicGeometry.Scheme.Hom.app AlgebraicGeometry.Spec map CommRingCat.ofHom CommRingCat.carrier CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PrimeSpectrum.Top CommRingCat.commRing TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder CommRingCat CommRingCat.instCategory CategoryTheory.ObjectProperty.FullSubcategory.obj CategoryTheory.Functor CategoryTheory.Functor.category CategoryTheory.Presheaf.IsSheaf Opens.grothendieckTopology structureSheaf Opposite.op AlgebraicGeometry.Scheme.Opens AlgebraicGeometry.PresheafedSpace.carrier AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom AlgebraicGeometry.Scheme.Hom.toLRSHom' AlgebraicGeometry.StructureSheaf.comap CommRingCat.Hom.hom le_rfl Set OmegaCompletePartialOrder.toPartialOrder CompleteLattice.instOmegaCompletePartialOrder CompleteBooleanAlgebra.toCompleteLattice CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra Set.instCompleteAtomicBooleanAlgebra TopologicalSpace.Opens.carrier	—	—
map_appLE 📖	mathematical	AlgebraicGeometry.Scheme.Opens AlgebraicGeometry.Spec Preorder.toLE PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder TopCat.carrier AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier CategoryTheory.Functor.obj Preorder.smallCategory TopologicalSpace.Opens TopCat.str TopologicalSpace.Opens.map AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom AlgebraicGeometry.Scheme.Hom.toLRSHom' map	AlgebraicGeometry.Scheme.Hom.appLE AlgebraicGeometry.Spec map CommRingCat.ofHom CommRingCat.carrier CategoryTheory.Functor.obj Opposite TopologicalSpace.Opens TopCat.carrier AlgebraicGeometry.PrimeSpectrum.Top CommRingCat.commRing TopCat.str CategoryTheory.Category.opposite Preorder.smallCategory PartialOrder.toPreorder TopologicalSpace.Opens.instPartialOrder CommRingCat CommRingCat.instCategory CategoryTheory.ObjectProperty.FullSubcategory.obj CategoryTheory.Functor CategoryTheory.Functor.category CategoryTheory.Presheaf.IsSheaf Opens.grothendieckTopology structureSheaf Opposite.op AlgebraicGeometry.StructureSheaf.comap CommRingCat.Hom.hom	—	—
map_apply 📖	mathematical	—	DFunLike.coe AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.Spec TopCat.carrier CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap TopCat.str ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom AlgebraicGeometry.Scheme.Hom.toLRSHom' map PrimeSpectrum.comap CommRingCat.carrier CommRing.toCommSemiring CommRingCat.commRing CommRingCat.Hom.hom	—	—
map_base 📖	mathematical	—	AlgebraicGeometry.PresheafedSpace.Hom.base CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.Spec AlgebraicGeometry.LocallyRingedSpace.Hom.toHom AlgebraicGeometry.Scheme.Hom.toLRSHom' map TopCat.ofHom PrimeSpectrum CommRingCat.carrier CommRing.toCommSemiring CommRingCat.commRing PrimeSpectrum.zariskiTopology PrimeSpectrum.comap CommRingCat.Hom.hom PrimeSpectrum.continuous_comap	—	—
map_base_apply 📖	mathematical	—	DFunLike.coe AlgebraicGeometry.PresheafedSpace.carrier CommRingCat CommRingCat.instCategory AlgebraicGeometry.SheafedSpace.toPresheafedSpace AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace AlgebraicGeometry.Scheme.toLocallyRingedSpace AlgebraicGeometry.Spec TopCat.carrier CategoryTheory.ConcreteCategory.hom TopCat TopCat.instCategory ContinuousMap TopCat.str ContinuousMap.instFunLike TopCat.instConcreteCategoryContinuousMapCarrier AlgebraicGeometry.PresheafedSpace.Hom.base AlgebraicGeometry.LocallyRingedSpace.Hom.toHom AlgebraicGeometry.Scheme.Hom.toLRSHom' map PrimeSpectrum.comap CommRingCat.carrier CommRing.toCommSemiring CommRingCat.commRing CommRingCat.Hom.hom	—	map_apply
map_comp 📖	mathematical	—	map CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory AlgebraicGeometry.Scheme AlgebraicGeometry.Scheme.instCategory AlgebraicGeometry.Spec	—	AlgebraicGeometry.Scheme.Hom.ext' locallyRingedSpaceMap_comp
map_comp_assoc 📖	mathematical	—	CategoryTheory.CategoryStruct.comp AlgebraicGeometry.Scheme CategoryTheory.Category.toCategoryStruct AlgebraicGeometry.Scheme.instCategory AlgebraicGeometry.Spec map CommRingCat CommRingCat.instCategory	—	CategoryTheory.Category.assoc Mathlib.Tactic.Reassoc.eq_whisker' map_comp
map_eqToHom 📖	mathematical	—	map CategoryTheory.eqToHom CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory AlgebraicGeometry.Scheme AlgebraicGeometry.Scheme.instCategory AlgebraicGeometry.Spec	—	map_id
map_id 📖	mathematical	—	map CategoryTheory.CategoryStruct.id CommRingCat CategoryTheory.Category.toCategoryStruct CommRingCat.instCategory AlgebraicGeometry.Scheme AlgebraicGeometry.Scheme.instCategory AlgebraicGeometry.Spec	—	AlgebraicGeometry.Scheme.Hom.ext' locallyRingedSpaceMap_id
map_inv 📖	mathematical	—	map CategoryTheory.inv CommRingCat CommRingCat.instCategory AlgebraicGeometry.Scheme AlgebraicGeometry.Scheme.instCategory AlgebraicGeometry.Spec AlgebraicGeometry.instIsIsoSchemeMapOfCommRingCat	—	AlgebraicGeometry.instIsIsoSchemeMapOfCommRingCat CategoryTheory.Functor.map_isIso CategoryTheory.isIso_op CategoryTheory.op_inv CategoryTheory.Functor.map_inv

CategoryTheory.Iso

Definitions

Name	Category	Theorems
schemeIsoToHomeo 📖	CompOp	—

SpecOfNotation

Definitions

Name	Category	Theorems
«termSpec(_)» 📖» "API Documentation")	CompOp	—

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