Basic

📁 Source: Mathlib/Algebra/Category/Ring/Basic.lean

Statistics

Metric	Count
Definitions commRingCatIsoToRingEquiv, commSemiRingCatIsoToRingEquiv, ringCatIsoToRingEquiv, semiRingCatIsoToRingEquiv, hom, hom, hom', carrier, commRing, forget_obj_eq_coe, fullyFaithfulForget₂ToRingCat, hasForgetToAddCommMonCat, hasForgetToRingCat, instCategory, instCoeSortType, instCommRingObjForgetRingHomCarrier, instConcreteCategoryRingHomCarrier, instHasForget₂RingHomCarrierCommMonCatMonoidHomCarrier, instInhabited, ofHom, CommSemiRingCat, hom, hom, hom', carrier, commSemiring, forget_obj_eq_coe, fullyFaithfulForget₂ToSemiRingCat, hasForgetToCommMonCat, hasForgetToSemiRingCat, instCategory, instCoeSortType, instCommSemiringObjForgetRingHomCarrier, instConcreteCategoryRingHomCarrier, instInhabited, ofHom, RingCat, hom, hom, hom', carrier, forget_obj_eq_coe, fullyFaithfulForget₂ToSemiRingCat, hasForgetToAddCommGrp, hasForgetToSemiRingCat, instCategory, instCoeSortType, instConcreteCategoryRingHomCarrier, instInhabited, instRingObjForgetRingHomCarrier, ofHom, toCommRingCatIso, toCommSemiRingCatIso, toRingCatIso, toSemiRingCatIso, SemiRingCat, hom, hom, hom', carrier, forget_obj_eq_coe, hasForgetToAddCommMonCat, hasForgetToMonCat, instCategory, instCoeSortType, instConcreteCategoryRingHomCarrier, instInhabited, instSemiringObjForgetRingHomCarrier, ofHom, semiring	70
Theorems commRingCatIsoToRingEquiv_toRingHom, commSemiRingCatIsoToRingEquiv_toRingHom, ringCatIsoToRingEquiv_toRingHom, semiRingCatIsoToRingEquiv_toRingHom, ext, ext_iff, coe_of, commMon_forget₂_map, commMon_forget₂_obj_coe, comp_apply, forgetReflectIsos, forgetToRingCat_map_hom, forgetToRingCat_obj, forget_map, forget_map_apply, forget_obj, hom_comp, hom_ext, hom_ext_iff, hom_id, hom_inv_apply, hom_ofHom, id_apply, instFullRingCatForget₂RingHomCarrierCarrier, inv_hom_apply, ofHom_apply, ofHom_comp, ofHom_hom, ofHom_id, of_carrier, ext, ext_iff, coe_of, comp_apply, forgetReflectIsos, forget_map, forget_obj, hom_comp, hom_ext, hom_ext_iff, hom_id, hom_inv_apply, hom_ofHom, id_apply, instFullSemiRingCatForget₂RingHomCarrierCarrier, inv_hom_apply, ofHom_apply, ofHom_comp, ofHom_hom, ofHom_id, of_carrier, ext, ext_iff, coe_of, comp_apply, forgetReflectIsos, forget_map, forget_map_apply, forget_obj, forget₂_map, hom_comp, hom_ext, hom_ext_iff, hom_id, hom_inv_apply, hom_ofHom, id_apply, instFullSemiRingCatForget₂RingHomCarrierCarrier, inv_hom_apply, ofHom_apply, ofHom_comp, ofHom_hom, ofHom_id, of_carrier, toCommRingCatIso_hom, toCommRingCatIso_inv, toCommSemiRingCatIso_hom, toCommSemiRingCatIso_inv, toRingCatIso_hom, toRingCatIso_inv, toSemiRingCatIso_hom, toSemiRingCatIso_inv, ext, ext_iff, coe_of, comp_apply, forgetReflectIsos, forget_map, forget_obj, forget₂_addCommMonCat_map, forget₂_monCat_map, hom_comp, hom_ext, hom_ext_iff, hom_id, hom_inv_apply, hom_ofHom, id_apply, inv_hom_apply, ofHom_apply, ofHom_comp, ofHom_hom, ofHom_id, of_carrier	104
Total	174

CategoryTheory.Iso

Definitions

Name	Category	Theorems
commRingCatIsoToRingEquiv 📖	CompOp	1 math math: commRingCatIsoToRingEquiv_toRingHom
commSemiRingCatIsoToRingEquiv 📖	CompOp	1 math math: commSemiRingCatIsoToRingEquiv_toRingHom
ringCatIsoToRingEquiv 📖	CompOp	1 math math: ringCatIsoToRingEquiv_toRingHom
semiRingCatIsoToRingEquiv 📖	CompOp	1 math math: semiRingCatIsoToRingEquiv_toRingHom

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
commRingCatIsoToRingEquiv_toRingHom 📖	mathematical	—	RingHomClass.toRingHom RingEquiv CommRingCat.carrier Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing CommRing.toNonUnitalCommRing CommRingCat.commRing Distrib.toAdd EquivLike.toFunLike RingEquiv.instEquivLike Semiring.toNonAssocSemiring CommSemiring.toSemiring CommRing.toCommSemiring RingEquivClass.toRingHomClass RingEquiv.instRingEquivClass commRingCatIsoToRingEquiv CommRingCat.Hom.hom hom CommRingCat CommRingCat.instCategory	—	RingEquivClass.toRingHomClass RingEquiv.instRingEquivClass
commSemiRingCatIsoToRingEquiv_toRingHom 📖	mathematical	—	RingHomClass.toRingHom RingEquiv CommSemiRingCat.carrier Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonAssocSemiring.toNonUnitalNonAssocSemiring Semiring.toNonAssocSemiring CommSemiring.toSemiring CommSemiRingCat.commSemiring Distrib.toAdd EquivLike.toFunLike RingEquiv.instEquivLike RingEquivClass.toRingHomClass RingEquiv.instRingEquivClass commSemiRingCatIsoToRingEquiv CommSemiRingCat.Hom.hom hom CommSemiRingCat CommSemiRingCat.instCategory	—	RingEquivClass.toRingHomClass RingEquiv.instRingEquivClass
ringCatIsoToRingEquiv_toRingHom 📖	mathematical	—	RingHomClass.toRingHom RingEquiv RingCat.carrier Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonAssocRing.toNonUnitalNonAssocRing Ring.toNonAssocRing RingCat.ring Distrib.toAdd EquivLike.toFunLike RingEquiv.instEquivLike Semiring.toNonAssocSemiring Ring.toSemiring RingEquivClass.toRingHomClass RingEquiv.instRingEquivClass ringCatIsoToRingEquiv RingCat.Hom.hom hom RingCat RingCat.instCategory	—	RingEquivClass.toRingHomClass RingEquiv.instRingEquivClass
semiRingCatIsoToRingEquiv_toRingHom 📖	mathematical	—	RingHomClass.toRingHom RingEquiv SemiRingCat.carrier Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonAssocSemiring.toNonUnitalNonAssocSemiring Semiring.toNonAssocSemiring SemiRingCat.semiring Distrib.toAdd EquivLike.toFunLike RingEquiv.instEquivLike RingEquivClass.toRingHomClass RingEquiv.instRingEquivClass semiRingCatIsoToRingEquiv SemiRingCat.Hom.hom hom SemiRingCat SemiRingCat.instCategory	—	RingEquivClass.toRingHomClass RingEquiv.instRingEquivClass

CommRingCat

Definitions

Name	Category	Theorems
carrier 📖	CompOp	721 math math: tensorProd_map_right, AlgebraicGeometry.Scheme.toSpecΓ_apply, PresheafOfModules.Monoidal.tensorObj_obj, AlgebraicGeometry.AffineSpace.map_Spec_map, smoothSheafCommRing.ι_forgetStalk_inv, AlgebraicGeometry.Scheme.fromSpecStalk_closedPoint, AlgebraicGeometry.Scheme.coprodPresheafObjIso_hom_fst_assoc, AlgebraicGeometry.Scheme.map_PrimeSpectrum_basicOpen_of_affine, forget₂Ring_preservesLimitsOfSize, KaehlerDifferential.map_d, AlgebraicGeometry.StructureSheaf.instIsScalarTowerCarrierStalkCommRingCatStructurePresheafInCommRingCatCarrierAbPresheafOpensCarrierTopModuleStructurePresheaf, AlgebraicGeometry.Scheme.zeroLocus_iInf, HomTopology.mvPolynomialHomeomorph_apply_snd, AlgebraicGeometry.IsAffineOpen.isLocalization_of_eq_basicOpen, AlgebraicGeometry.Etale.etale_appLE, AlgebraicGeometry.LocallyQuasiFinite.quasiFinite_appLE, AlgebraicGeometry.ProjectiveSpectrum.Proj.awayToSection_apply, instFullRingCatForget₂RingHomCarrierCarrier, instNontrivialCarrierStalkCommRingCatPresheafSmoothSheafCommRing, AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction_counit_app, forget_preservesLimitsOfSize, forgetReflectIsos, AlgebraicGeometry.RingedSpace.basicOpen_res, AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv, closure_range_union_range_eq_top_of_isPushout, Localization.epi', AlgebraicGeometry.LocallyOfFiniteType.finiteType_of_affine_subset, AlgebraicGeometry.Scheme.zeroLocus_empty_eq_univ, AlgebraicGeometry.surjectiveOnStalks_iff, AlgebraicGeometry.instIsDomainCarrierStalkCommRingCatPresheafOfIsIntegral, smoothSheafCommRing.nonunits_stalk, AlgebraicGeometry.IsLocallyArtinian.isArtinianRing_presheaf_obj, AlgebraicGeometry.LocallyRingedSpace.notMem_prime_iff_unit_in_stalk, AlgebraicGeometry.instIsDomainCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTopOfIsIntegral, AlgebraicGeometry.StructureSheaf.comap_comp, AlgebraicGeometry.AffineSpace.toSpecMvPolyIntEquiv_symm_apply, AlgebraicGeometry.functionField_isFractionRing_of_isAffineOpen, AlgebraicGeometry.Scheme.IdealSheafData.ideal_sSup, AlgebraicGeometry.StructureSheaf.IsLocalization.to_stalk, AlgebraicGeometry.AffineSpace.SpecIso_hom_appTop, AlgebraicGeometry.LocallyRingedSpace.basicOpen_eq_bot_iff_forall_evaluation_eq_zero, AlgebraicGeometry.isIso_SpecMap_iff, AlgebraicGeometry.Spec.locallyRingedSpaceObj_presheaf_map, AlgebraicGeometry.Scheme.mem_basicOpen', AlgebraicGeometry.Scheme.zeroLocus_eq_univ_iff_subset_nilradical, AlgebraicGeometry.ProjectiveSpectrum.Proj.toSpec_preimage_basicOpen, AlgebraicGeometry.Proj.pow_apply, id_apply, AlgebraicGeometry.pointsPi_surjective_of_isAffine, AlgebraicGeometry.Spec_presheaf, AlgebraicGeometry.HasRingHomProperty.stalkMap, AlgebraicGeometry.HasAffineProperty.SpecMap_iff_of_affineAnd, AlgebraicGeometry.StructureSheaf.toOpen_comp_comap_apply, smoothSheafCommRing.ι_evalHom_apply, AlgebraicGeometry.ι_sigmaSpec, AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec_hom_apply_asIdeal, AlgebraicGeometry.Scheme.zeroLocus_mul, AlgebraicGeometry.Proj.mul_apply, AlgebraicGeometry.StructureSheaf.comap_id', AlgebraicGeometry.instIsIsoSchemeSigmaSpecOfFinite, AlgebraicGeometry.SpecMap_preimage_basicOpen, AlgebraicGeometry.isField_stalk_of_closure_mem_irreducibleComponents, mkUnder_ext_iff, free_obj_coe, AlgebraicGeometry.Scheme.ideal_ker_le_ker_ΓSpecIso_inv_comp, smoothSheafCommRing.eval_germ, moduleCatRestrictScalarsPseudofunctor_mapComp, AlgebraicGeometry.pointsPi_injective, AlgebraicGeometry.Scheme.basic_open_isOpenImmersion, AlgebraicGeometry.isIso_ΓSpec_adjunction_unit_app_basicOpen, forget₂CommSemiRing_preservesLimitsOfSize, AlgebraicGeometry.isNoetherian_iff_of_finite_affine_openCover, AlgebraicGeometry.localRingHom_comp_stalkIso_apply, AlgebraicGeometry.instIsLeftAdjointModuleCatCarrierModulesSpecOfFunctor, AlgebraicGeometry.functionField_isFractionRing_of_affine, AlgebraicGeometry.tilde.map_id, CommAlgCat.forget₂_commRingCat_obj, AlgebraicGeometry.IsAffineOpen.fromSpecStalk_closedPoint, AlgebraicGeometry.SmoothOfRelativeDimension.exists_isStandardSmoothOfRelativeDimension, AlgebraicGeometry.Scheme.IdealSheafData.ideal_le_comap_ideal, forget_obj, PresheafOfModules.Derivation.d_map, AlgebraicGeometry.Scheme.IdealSheafData.equivOfIsAffine_apply, AlgebraicGeometry.ValuativeCriterion.Existence.instIsLocalHomCarrierRingHomHomHomCommRingCat, free_map_coe, AlgebraicGeometry.StructureSheaf.instAtPrimeCarrierStalkCommRingCatStructurePresheafInCommRingCatAsIdeal, AlgebraicGeometry.tilde.map_sub, toAlgHom_comp, AlgebraicGeometry.Scheme.IdealSheafData.equivOfIsAffine_symm_apply, AlgebraicGeometry.Scheme.IdealSheafData.ideal_mul, AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.structurePresheafInCommRing_obj_carrier, smoothSheafCommRing.instLocalRing_stalk, AlgebraicGeometry.locallyQuasiFinite_iff, AlgebraicGeometry.StructureSheaf.toOpen_comp_comap, AlgebraicGeometry.affineLocally_iff_forall_isAffineOpen, AlgebraicGeometry.Scheme.Hom.formallyUnramified_appLE, AlgebraicGeometry.Scheme.isoSpec_inv_preimage_zeroLocus, coyoneda_map_app, AlgebraicGeometry.functionField_isScalarTower, AlgebraicGeometry.Scheme.Modules.toOpen_fromTildeΓ_app, AlgebraicGeometry.Scheme.exists_germ_injective, AlgebraicGeometry.LocallyRingedSpace.toΓSpecSheafedSpace_app_eq, AlgebraicGeometry.Scheme.coprodPresheafObjIso_hom_snd_assoc, AlgebraicGeometry.Scheme.Hom.toNormalization_app_preimage, AlgebraicGeometry.Proj.one_apply, AlgebraicGeometry.Scheme.Hom.toImage_app_injective, AlgebraicGeometry.StructureSheaf.to_basicOpen_epi, AlgebraicGeometry.Scheme.IdealSheafData.ideal_le_ker_glueDataObjι, AlgebraicGeometry.Scheme.isOpenImmersion_SpecMap_localizationAway, AlgebraicGeometry.SpecToEquivOfLocalRing_apply_snd_coe, AlgebraicGeometry.IsAffineOpen.self_le_iSup_basicOpen_iff, AlgebraicGeometry.Scheme.zeroLocus_biInf, AlgebraicGeometry.Scheme.AffineZariskiSite.generate_presieveOfSections, AlgebraicGeometry.instAdditiveModuleCatCarrierModulesSpecOfFunctor, coyoneda_obj_obj_carrier, AlgebraicGeometry.SurjectiveOnStalks.iff_of_isAffine, AlgebraicGeometry.IsAffineOpen.isLocalization_basicOpen, AlgebraicGeometry.Scheme.zeroLocus_iUnion, AlgebraicGeometry.IsReduced.component_reduced, AlgebraicGeometry.stalkwise_SpecMap_iff, coyoneda_obj_map, AlgebraicGeometry.IsIntegralHom.isIntegral_app, AlgebraicGeometry.Scheme.ker_ideal_of_isPullback_of_isOpenImmersion, PresheafOfModules.Derivation.postcomp_d_apply, ofHom_hom, AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv_appTop_coord, PresheafOfModules.Derivation.d_one, AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv_over_assoc, AlgebraicGeometry.ProjectiveSpectrum.Proj.awayToΓ_ΓToStalk, AlgebraicGeometry.StructureSheaf.comap_const, AlgebraicGeometry.Scheme.zeroLocus_singleton, AlgebraicGeometry.LocallyRingedSpace.component_nontrivial, AlgebraicGeometry.Scheme.IdealSheafData.ideal_map, AlgebraicGeometry.Proj.sub_apply, AlgebraicGeometry.isLocallyArtinian_iff_of_isOpenCover, PresheafOfModules.Derivation.d_mul, pushout_inr_tensorProdObjIsoPushoutObj_inv_right_assoc, AlgebraicGeometry.Scheme.Hom.ι_toNormalization, AlgebraicGeometry.StructureSheaf.comap_apply, AlgebraicGeometry.Scheme.Hom.smooth_appLE, AlgebraicGeometry.Scheme.IdealSheafData.ideal_pow, AlgebraicGeometry.RingedSpace.mem_top_basicOpen, AlgebraicGeometry.IsAffineOpen.ideal_le_iff, AlgebraicGeometry.stalkToFiberRingHom_homogeneousLocalizationToStalk, AlgebraicGeometry.Scheme.Hom.QuasiFiniteAt.quasiFiniteAt, AlgebraicGeometry.isLocallyNoetherian_iff_of_affine_openCover, AlgebraicGeometry.Scheme.IdealSheafData.mem_support_iff_of_mem, AlgHom.toUnder_comp, AlgebraicGeometry.IsLocallyArtinian.isArtinianRing_of_isAffine, AlgebraicGeometry.StructureSheaf.toPushforwardStalkAlgHom_apply, AlgebraicGeometry.Scheme.Hom.germ_stalkMap_apply, AlgebraicGeometry.Scheme.preimage_basicOpen_top, AlgebraicGeometry.Scheme.ΓSpecIso_inv, AlgebraicGeometry.IsAffineOpen.basicOpen_fromSpec_app, RingHom.RespectsIso.cancel_right_isIso, AlgebraicGeometry.Scheme.isoSpec_inv_image_zeroLocus, AlgebraicGeometry.Scheme.IdealSheafData.range_glueDataObjι_ι, AlgebraicGeometry.Scheme.germToFunctionField_injective, moduleCatRestrictScalarsPseudofunctor_mapId, instIsRightAdjointCommMonCatForget₂RingHomCarrierMonoidHomCarrier, AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction_counit_app', AlgebraicGeometry.instFullModuleCatCarrierModulesSpecOfFunctor, AlgebraicGeometry.Scheme.Opens.ι_image_basicOpen_topIso_inv, PresheafOfModules.Monoidal.tensorObj_map_tmul, AlgebraicGeometry.Scheme.zeroLocus_setMul, AlgebraicGeometry.Scheme.toSpecΓ_preimage_zeroLocus, commAlgCatEquivUnder_functor_obj, TopCat.Presheaf.stalk_open_algebraMap, AlgebraicGeometry.Scheme.Hom.app_surjective, smoothSheafCommRing.forgetStalk_inv_comp_eval_assoc, AlgebraicGeometry.affineLocally_iff_affineOpens_le, AlgebraicGeometry.RingedSpace.basicOpen_pow, AlgebraicGeometry.HasRingHomProperty.iff_of_iSup_eq_top, AlgebraicGeometry.Scheme.IdealSheafData.vanishingIdeal_ideal, coproductCocone_inl, AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff', AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.invApp_app_apply, AlgebraicGeometry.IsPreimmersion.Spec_map_iff, smoothSheafCommRing.forgetStalk_inv_comp_eval, AlgebraicGeometry.IsAffineOpen.primeIdealOf_isMaximal_of_isClosed, AlgebraicGeometry.Scheme.IdealSheafData.ideal_bot, AlgebraicGeometry.Scheme.ker_of_isAffine, AlgebraicGeometry.IsAffineOpen.algebraMap_Spec_obj, AlgebraicGeometry.isBasis_basicOpen, AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff, AlgebraicGeometry.LocallyRingedSpace.isLocalHomStalkMap, AlgebraicGeometry.Scheme.Γevaluation_naturality_apply, AlgebraicGeometry.locallyOfFiniteType_iff, AlgebraicGeometry.Flat.flat_and_surjective_iff_faithfullyFlat_of_isAffine, ModuleCat.Derivation.d_mul, AlgebraicGeometry.LocallyRingedSpace.coe_toΓSpecSheafedSpace_hom_base_hom_apply_asIdeal, AlgebraicGeometry.Scheme.Opens.toSpecΓ_preimage_basicOpen, smoothSheafCommRing.forgetStalk_inv_comp_eval_apply, AlgebraicGeometry.isIntegral_appTop_of_universallyClosed, AlgebraicGeometry.Scheme.zeroLocus_def, AlgebraicGeometry.LocallyRingedSpace.toΓSpec_continuous, AlgebraicGeometry.Spec_map_localization_isIso, AlgebraicGeometry.StructureSheaf.isLocalizedModule_toPushforwardStalkAlgHom_aux, AlgebraicGeometry.Scheme.stalkClosedPointTo_fromSpecStalk, smoothSheafCommRing.isUnit_stalk_iff, AlgebraicGeometry.tilde.isoTop_hom, AlgebraicGeometry.ProjectiveSpectrum.Proj.isLocalization_atPrime, ModuleCat.Tilde.toOpen_res, AlgebraicGeometry.Scheme.IdealSheafData.mem_supportSet_iff, AlgebraicGeometry.Scheme.IdealSheafData.mem_support_iff, AlgebraicGeometry.LocallyOfFiniteType.finiteType_appLE, mkUnder_hom, AlgebraicGeometry.Spec.map_appLE, AlgebraicGeometry.Scheme.Hom.finitePresentation_appLE, AlgebraicGeometry.Γ_restrict_isLocalization, AlgEquiv.toUnder_trans, AlgebraicGeometry.RingedSpace.res_zero, AlgebraicGeometry.affine_isIntegral_iff, instIsLocalHomCarrierObjWalkingParallelPairFunctorConstPtEqualizerForkZeroParallelPairRingHomHomι, moduleCatRestrictScalarsPseudofunctor_map, coproductCocone_inr, subsingleton_of_isTerminal, AlgebraicGeometry.Spec.germ_stalkMapIso_hom_assoc, AlgebraicGeometry.Scheme.IdealSheafData.le_ofIdeals_iff, AlgebraicGeometry.Scheme.Hom.preimage_basicOpen_top, AlgebraicGeometry.FormallyUnramified.formallyUnramified_of_affine_subset, forget_map, AlgebraicGeometry.StructureSheaf.IsLocalization.to_basicOpen, AlgebraicGeometry.formallySmooth_stalkMap_iff, TopCat.Sheaf.objSupIsoProdEqLocus_hom_fst, AlgebraicGeometry.AffineSpace.spec_le_iff, smoothSheafCommRing.evalHom_germ, AlgebraicGeometry.LocallyRingedSpace.residue_comp_residueFieldMap_eq_stalkMap_comp_residue, AlgebraicGeometry.HasRingHomProperty.iff_appLE, PresheafOfModules.Derivation.congr_d, AlgebraicGeometry.StructureSheaf.toOpen_comp_comap_assoc, AlgebraicGeometry.IsAffineOpen.self_le_basicOpen_union_iff, AlgebraicGeometry.exists_appTop_π_eq_of_isAffine_of_isLimit, Under.tensorProdEqualizer_ι, AlgebraicGeometry.tilde.functor_map, AlgebraicGeometry.Scheme.IdealSheafData.coe_support_ofIdealTop, AlgebraicGeometry.Scheme.component_nontrivial, AlgebraicGeometry.Scheme.zeroLocus_iInf_of_nonempty, AlgebraicGeometry.Scheme.AffineZariskiSite.generate_presieveOfSections_mem_grothendieckTopology, AlgebraicGeometry.ΓSpec.toOpen_comp_locallyRingedSpaceAdjunction_homEquiv_app, AlgebraicGeometry.Scheme.basicOpen_pow, AlgebraicGeometry.HasRingHomProperty.appLE, AlgebraicGeometry.Scheme.Hom.ker_apply, AlgebraicGeometry.Scheme.IdealSheafData.ideal_comap_of_isOpenImmersion, AlgebraicGeometry.AffineSpace.SpecIso_inv_appTop_coord, AlgebraicGeometry.germ_stalkClosedPointIso_hom_assoc, toAlgHom_id, smoothSheafCommRing.forgetStalk_hom_comp_evalHom_assoc, AlgebraicGeometry.Scheme.evaluation_eq_zero_iff_notMem_basicOpen, AlgebraicGeometry.LocallyRingedSpace.restrictStalkIso_inv_eq_germ_apply, AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff_of_mem_mul, hom_comp, AlgebraicGeometry.LocallyRingedSpace.stalkMap_inv_hom_apply, AlgebraicGeometry.Scheme.stalkSpecializes_stalkMap_apply, AlgebraicGeometry.Scheme.zeroLocus_radical, AlgebraicGeometry.Scheme.AffineZariskiSite.presieveOfSections_surjective, HomTopology.isClosedEmbedding_hom, AlgebraicGeometry.StructureSheaf.globalSectionsIso_hom, AlgebraicGeometry.Scheme.Spec.residue_residueFieldIso_hom_assoc, AlgebraicGeometry.tilde.map_comp_assoc, AlgebraicGeometry.Scheme.IdealSheafData.map_ideal, hom_inv_apply, AlgebraicGeometry.Scheme.basicOpen_add_le, AlgebraicGeometry.Scheme.zeroLocus_map, AlgebraicGeometry.Scheme.Hom.finite_app, AlgebraicGeometry.Scheme.AffineZariskiSite.coequifibered_iff_forall_isLocalizationAway, AlgebraicGeometry.IsAffineOpen.fromSpec_preimage_zeroLocus, AlgebraicGeometry.Scheme.Modules.toOpen_fromTildeΓ_app_assoc, AlgebraicGeometry.FormallyUnramified.instIsSeparableCarrierResidueFieldCoeContinuousMapCarrierCarrierCommRingCatHomTopCatBaseOfLocallyOfFiniteType, AlgebraicGeometry.Scheme.Hom.ideal_ker_le, AlgebraicGeometry.AffineSpace.comp_homOfVector, AlgebraicGeometry.smooth_iff, AlgebraicGeometry.Scheme.zeroLocus_inf, AlgebraicGeometry.Scheme.IdealSheafData.ofIdeals_mono, AlgebraicGeometry.IsClosedImmersion.spec_of_quotient_mk, AlgebraicGeometry.germ_injective_of_isIntegral, ModuleCat.Derivation.desc_d, AlgebraicGeometry.isClosedImmersion_iff_isAffineHom, AlgebraicGeometry.AffineSpace.hom_ext_iff, forget₂CommSemiRing_preservesLimits, AlgebraicGeometry.ΓSpec.unop_locallyRingedSpaceAdjunction_counit_app', AlgebraicGeometry.Scheme.Hom.preimage_basicOpen, AlgebraicGeometry.stalkMap_toStalk, AlgebraicGeometry.Scheme.IdealSheafData.subschemeι_app, AlgEquiv.toUnder_inv_right_apply, HomTopology.mvPolynomialHomeomorph_symm_apply_hom, AlgebraicGeometry.Scheme.basicOpen_res_eq, AlgebraicGeometry.StructureSheaf.isLocalizedModule_toPushforwardStalkAlgHom, AlgebraicGeometry.Scheme.Hom.stalkFunctor_toImage_injective, AlgebraicGeometry.isNoetherian_iff_of_finite_iSup_eq_top, AlgebraicGeometry.Scheme.image_basicOpen, AlgebraicGeometry.tilde.isIso_toOpen_top, AlgebraicGeometry.LocallyRingedSpace.Hom.prop, TopCat.Presheaf.SubmonoidPresheaf.map, AlgebraicGeometry.SurjectiveOnStalks.surj_on_stalks, AlgebraicGeometry.Spec.fromSpecStalk_eq', AlgebraicGeometry.HasRingHomProperty.appTop, AlgebraicGeometry.LocallyRingedSpace.basicOpen_zero, commMon_forget₂_obj_coe, AlgebraicGeometry.isLocallyNoetherian_Spec, toAlgHom_apply, coe_of, AlgebraicGeometry.Scheme.isNilpotent_iff_basicOpen_eq_bot, AlgebraicGeometry.SurjectiveOnStalks.stalkMap_surjective, AlgebraicGeometry.IsIntegral.component_integral, pushout_inl_tensorProdObjIsoPushoutObj_inv_right_assoc, HomTopology.isEmbedding_hom, AlgebraicGeometry.Scheme.instSubsingletonCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensBot, AlgebraicGeometry.LocallyRingedSpace.restrictStalkIso_hom_eq_germ_apply, moduleCatExtendScalarsPseudofunctor_map, smoothSheafCommRing.ι_forgetStalk_inv_apply, AlgebraicGeometry.instIsFractionRingCarrierStalkCommRingCatPresheafFunctionField, AlgebraicGeometry.germ_stalkClosedPointIso_hom, AlgebraicGeometry.IsFinite.instHasAffinePropertyAndIsAffineFiniteCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTopHomAppTop, AlgebraicGeometry.SpecToEquivOfLocalRing_apply_fst, coproductCocone_pt, monoidAlgebra_map, AlgebraicGeometry.tilde.instIsLocalizedModuleCarrierCarrierOfCarrierStalkAbPresheafPrimeComplAsIdealHomToStalk, AlgebraicGeometry.Spec.map_apply, forgetToRingCat_obj, Colimits.cocone_naturality_components, AlgebraicGeometry.Spec.sheafedSpaceMap_hom_c_app, AlgebraicGeometry.Scheme.evaluation_naturality_apply, smoothSheafCommRing.eval_surjective, AlgebraicGeometry.Scheme.residue_surjective, AlgebraicGeometry.IsAffineOpen.iSup_basicOpen_eq_self_iff, AlgebraicGeometry.Scheme.basicOpen_one, AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff_of_mem, AlgebraicGeometry.targetAffineLocally_affineAnd_iff, smoothSheafCommRing.forgetStalk_hom_comp_evalHom, AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_stalk_isLocalHom, AlgebraicGeometry.Scheme.zeroLocus_eq_univ_iff_subset_nilradical_of_isCompact, FilteredColimits.forget_preservesFilteredColimits, AlgebraicGeometry.LocallyRingedSpace.Γevaluation_naturality_apply, moduleCatExtendScalarsPseudofunctor_obj, AlgebraicGeometry.StructureSheaf.toStalk_stalkSpecializes_apply, commAlgCatEquivUnder_counitIso, AlgebraicGeometry.Spec_zeroLocus_eq_zeroLocus, AlgebraicGeometry.Scheme.coprodPresheafObjIso_hom_snd, RingHom.surjective_iff_epi_and_finite, AlgebraicGeometry.Scheme.map_basicOpen, AlgebraicGeometry.Proj.zero_apply, PresheafOfModules.Derivation'.d_app, AlgebraicGeometry.Spec_carrier, TopCat.Sheaf.objSupIsoProdEqLocus_inv_snd, smoothSheafCommRing.ι_forgetStalk_inv_assoc, AlgebraicGeometry.IsAffineOpen.basicOpenSectionsToAffine_isIso, pushout_inl_tensorProdObjIsoPushoutObj_inv_right, AlgebraicGeometry.exists_smooth_of_formallySmooth_stalk, AlgebraicGeometry.Scheme.IdealSheafData.ideal_top, commAlgCatEquivUnder_inverse_obj_carrier, TopCat.Presheaf.instIsLocalizationCarrierObjOppositeOpensCarrierCommRingCatObjLocalizationPresheaf, AlgebraicGeometry.LocallyOfFiniteType.stalkMap, ModuleCat.Derivation.d_map, AlgebraicGeometry.Scheme.Hom.etale_appLE, AlgebraicGeometry.AffineSpace.map_SpecMap, AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalHom, ofHom_apply, AlgebraicGeometry.Scheme.IdealSheafData.isLocalization_away, RingHom.RespectsIso.cancel_left_isIso, AlgebraicGeometry.IsClosedImmersion.hasAffineProperty, AlgebraicGeometry.Scheme.Spec.algebraMap_residueFieldIso_inv, AlgHom.toUnder_right, 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AlgebraicGeometry.Proj.mul_apply, AlgebraicGeometry.StructureSheaf.comap_id', AlgebraicGeometry.instIsIsoSchemeSigmaSpecOfFinite, AlgebraicGeometry.SpecMap_preimage_basicOpen, AlgebraicGeometry.isField_stalk_of_closure_mem_irreducibleComponents, mkUnder_ext_iff, AlgebraicGeometry.Scheme.ideal_ker_le_ker_ΓSpecIso_inv_comp, smoothSheafCommRing.eval_germ, moduleCatRestrictScalarsPseudofunctor_mapComp, AlgebraicGeometry.pointsPi_injective, AlgebraicGeometry.Scheme.basic_open_isOpenImmersion, AlgebraicGeometry.isIso_ΓSpec_adjunction_unit_app_basicOpen, forget₂CommSemiRing_preservesLimitsOfSize, AlgebraicGeometry.isNoetherian_iff_of_finite_affine_openCover, AlgebraicGeometry.localRingHom_comp_stalkIso_apply, AlgebraicGeometry.instIsLeftAdjointModuleCatCarrierModulesSpecOfFunctor, AlgebraicGeometry.functionField_isFractionRing_of_affine, AlgebraicGeometry.tilde.map_id, CommAlgCat.forget₂_commRingCat_obj, AlgebraicGeometry.IsAffineOpen.fromSpecStalk_closedPoint, AlgebraicGeometry.SmoothOfRelativeDimension.exists_isStandardSmoothOfRelativeDimension, AlgebraicGeometry.Scheme.IdealSheafData.ideal_le_comap_ideal, forget_obj, PresheafOfModules.Derivation.d_map, AlgebraicGeometry.Scheme.IdealSheafData.equivOfIsAffine_apply, AlgebraicGeometry.ValuativeCriterion.Existence.instIsLocalHomCarrierRingHomHomHomCommRingCat, free_map_coe, AlgebraicGeometry.StructureSheaf.instAtPrimeCarrierStalkCommRingCatStructurePresheafInCommRingCatAsIdeal, AlgebraicGeometry.tilde.map_sub, toAlgHom_comp, AlgebraicGeometry.Scheme.IdealSheafData.equivOfIsAffine_symm_apply, AlgebraicGeometry.Scheme.IdealSheafData.ideal_mul, smoothSheafCommRing.instLocalRing_stalk, AlgebraicGeometry.locallyQuasiFinite_iff, AlgebraicGeometry.StructureSheaf.toOpen_comp_comap, AlgebraicGeometry.affineLocally_iff_forall_isAffineOpen, AlgebraicGeometry.Scheme.Hom.formallyUnramified_appLE, AlgebraicGeometry.Scheme.isoSpec_inv_preimage_zeroLocus, coyoneda_map_app, AlgebraicGeometry.functionField_isScalarTower, AlgebraicGeometry.Scheme.Modules.toOpen_fromTildeΓ_app, AlgebraicGeometry.Scheme.exists_germ_injective, AlgebraicGeometry.LocallyRingedSpace.toΓSpecSheafedSpace_app_eq, AlgebraicGeometry.Scheme.coprodPresheafObjIso_hom_snd_assoc, AlgebraicGeometry.Scheme.Hom.toNormalization_app_preimage, AlgebraicGeometry.Proj.one_apply, AlgebraicGeometry.Scheme.Hom.toImage_app_injective, AlgebraicGeometry.StructureSheaf.to_basicOpen_epi, AlgebraicGeometry.Scheme.IdealSheafData.ideal_le_ker_glueDataObjι, AlgebraicGeometry.Scheme.isOpenImmersion_SpecMap_localizationAway, AlgebraicGeometry.SpecToEquivOfLocalRing_apply_snd_coe, AlgebraicGeometry.IsAffineOpen.self_le_iSup_basicOpen_iff, AlgebraicGeometry.Scheme.zeroLocus_biInf, AlgebraicGeometry.Scheme.AffineZariskiSite.generate_presieveOfSections, AlgebraicGeometry.instAdditiveModuleCatCarrierModulesSpecOfFunctor, AlgebraicGeometry.SurjectiveOnStalks.iff_of_isAffine, AlgebraicGeometry.IsAffineOpen.isLocalization_basicOpen, AlgebraicGeometry.IsReduced.component_reduced, AlgebraicGeometry.stalkwise_SpecMap_iff, coyoneda_obj_map, AlgebraicGeometry.IsIntegralHom.isIntegral_app, AlgebraicGeometry.Scheme.ker_ideal_of_isPullback_of_isOpenImmersion, PresheafOfModules.Derivation.postcomp_d_apply, ofHom_hom, AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv_appTop_coord, PresheafOfModules.Derivation.d_one, AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv_over_assoc, AlgebraicGeometry.ProjectiveSpectrum.Proj.awayToΓ_ΓToStalk, AlgebraicGeometry.StructureSheaf.comap_const, AlgebraicGeometry.Scheme.IdealSheafData.ideal_map, AlgebraicGeometry.Proj.sub_apply, AlgebraicGeometry.isLocallyArtinian_iff_of_isOpenCover, PresheafOfModules.Derivation.d_mul, pushout_inr_tensorProdObjIsoPushoutObj_inv_right_assoc, AlgebraicGeometry.Scheme.Hom.ι_toNormalization, AlgebraicGeometry.StructureSheaf.comap_apply, AlgebraicGeometry.Scheme.Hom.smooth_appLE, AlgebraicGeometry.Scheme.IdealSheafData.ideal_pow, AlgebraicGeometry.RingedSpace.mem_top_basicOpen, AlgebraicGeometry.IsAffineOpen.ideal_le_iff, AlgebraicGeometry.stalkToFiberRingHom_homogeneousLocalizationToStalk, AlgebraicGeometry.Scheme.Hom.QuasiFiniteAt.quasiFiniteAt, AlgebraicGeometry.isLocallyNoetherian_iff_of_affine_openCover, AlgebraicGeometry.Scheme.IdealSheafData.mem_support_iff_of_mem, AlgHom.toUnder_comp, AlgebraicGeometry.IsLocallyArtinian.isArtinianRing_of_isAffine, AlgebraicGeometry.StructureSheaf.toPushforwardStalkAlgHom_apply, AlgebraicGeometry.Scheme.Hom.germ_stalkMap_apply, AlgebraicGeometry.Scheme.preimage_basicOpen_top, AlgebraicGeometry.Scheme.ΓSpecIso_inv, AlgebraicGeometry.IsAffineOpen.basicOpen_fromSpec_app, RingHom.RespectsIso.cancel_right_isIso, AlgebraicGeometry.Scheme.isoSpec_inv_image_zeroLocus, AlgebraicGeometry.Scheme.IdealSheafData.range_glueDataObjι_ι, AlgebraicGeometry.Scheme.germToFunctionField_injective, moduleCatRestrictScalarsPseudofunctor_mapId, instIsRightAdjointCommMonCatForget₂RingHomCarrierMonoidHomCarrier, AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction_counit_app', AlgebraicGeometry.instFullModuleCatCarrierModulesSpecOfFunctor, AlgebraicGeometry.Scheme.Opens.ι_image_basicOpen_topIso_inv, PresheafOfModules.Monoidal.tensorObj_map_tmul, AlgebraicGeometry.Scheme.zeroLocus_setMul, AlgebraicGeometry.Scheme.toSpecΓ_preimage_zeroLocus, commAlgCatEquivUnder_functor_obj, TopCat.Presheaf.stalk_open_algebraMap, AlgebraicGeometry.Scheme.Hom.app_surjective, smoothSheafCommRing.forgetStalk_inv_comp_eval_assoc, AlgebraicGeometry.affineLocally_iff_affineOpens_le, AlgebraicGeometry.RingedSpace.basicOpen_pow, AlgebraicGeometry.HasRingHomProperty.iff_of_iSup_eq_top, AlgebraicGeometry.Scheme.IdealSheafData.vanishingIdeal_ideal, coproductCocone_inl, AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff', AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.invApp_app_apply, AlgebraicGeometry.IsPreimmersion.Spec_map_iff, smoothSheafCommRing.forgetStalk_inv_comp_eval, AlgebraicGeometry.IsAffineOpen.primeIdealOf_isMaximal_of_isClosed, AlgebraicGeometry.Scheme.IdealSheafData.ideal_bot, AlgebraicGeometry.Scheme.ker_of_isAffine, AlgebraicGeometry.IsAffineOpen.algebraMap_Spec_obj, AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff, AlgebraicGeometry.LocallyRingedSpace.isLocalHomStalkMap, AlgebraicGeometry.Scheme.Γevaluation_naturality_apply, AlgebraicGeometry.locallyOfFiniteType_iff, AlgebraicGeometry.Flat.flat_and_surjective_iff_faithfullyFlat_of_isAffine, ModuleCat.Derivation.d_mul, AlgebraicGeometry.LocallyRingedSpace.coe_toΓSpecSheafedSpace_hom_base_hom_apply_asIdeal, AlgebraicGeometry.Scheme.Opens.toSpecΓ_preimage_basicOpen, smoothSheafCommRing.forgetStalk_inv_comp_eval_apply, AlgebraicGeometry.isIntegral_appTop_of_universallyClosed, AlgebraicGeometry.LocallyRingedSpace.toΓSpec_continuous, AlgebraicGeometry.Spec_map_localization_isIso, AlgebraicGeometry.StructureSheaf.isLocalizedModule_toPushforwardStalkAlgHom_aux, AlgebraicGeometry.Scheme.stalkClosedPointTo_fromSpecStalk, smoothSheafCommRing.isUnit_stalk_iff, AlgebraicGeometry.tilde.isoTop_hom, AlgebraicGeometry.ProjectiveSpectrum.Proj.isLocalization_atPrime, ModuleCat.Tilde.toOpen_res, AlgebraicGeometry.Scheme.IdealSheafData.mem_supportSet_iff, AlgebraicGeometry.Scheme.IdealSheafData.mem_support_iff, AlgebraicGeometry.LocallyOfFiniteType.finiteType_appLE, mkUnder_hom, AlgebraicGeometry.Spec.map_appLE, AlgebraicGeometry.Scheme.Hom.finitePresentation_appLE, AlgebraicGeometry.Γ_restrict_isLocalization, AlgEquiv.toUnder_trans, AlgebraicGeometry.RingedSpace.res_zero, AlgebraicGeometry.affine_isIntegral_iff, instIsLocalHomCarrierObjWalkingParallelPairFunctorConstPtEqualizerForkZeroParallelPairRingHomHomι, moduleCatRestrictScalarsPseudofunctor_map, coproductCocone_inr, AlgebraicGeometry.Spec.germ_stalkMapIso_hom_assoc, AlgebraicGeometry.Scheme.IdealSheafData.le_ofIdeals_iff, AlgebraicGeometry.Scheme.Hom.preimage_basicOpen_top, AlgebraicGeometry.FormallyUnramified.formallyUnramified_of_affine_subset, forget_map, AlgebraicGeometry.StructureSheaf.IsLocalization.to_basicOpen, AlgebraicGeometry.formallySmooth_stalkMap_iff, TopCat.Sheaf.objSupIsoProdEqLocus_hom_fst, AlgebraicGeometry.AffineSpace.spec_le_iff, smoothSheafCommRing.evalHom_germ, AlgebraicGeometry.LocallyRingedSpace.residue_comp_residueFieldMap_eq_stalkMap_comp_residue, AlgebraicGeometry.HasRingHomProperty.iff_appLE, PresheafOfModules.Derivation.congr_d, AlgebraicGeometry.StructureSheaf.toOpen_comp_comap_assoc, AlgebraicGeometry.IsAffineOpen.self_le_basicOpen_union_iff, AlgebraicGeometry.exists_appTop_π_eq_of_isAffine_of_isLimit, Under.tensorProdEqualizer_ι, AlgebraicGeometry.tilde.functor_map, AlgebraicGeometry.Scheme.IdealSheafData.coe_support_ofIdealTop, AlgebraicGeometry.Scheme.zeroLocus_iInf_of_nonempty, AlgebraicGeometry.Scheme.AffineZariskiSite.generate_presieveOfSections_mem_grothendieckTopology, AlgebraicGeometry.ΓSpec.toOpen_comp_locallyRingedSpaceAdjunction_homEquiv_app, AlgebraicGeometry.Scheme.basicOpen_pow, AlgebraicGeometry.HasRingHomProperty.appLE, AlgebraicGeometry.Scheme.Hom.ker_apply, AlgebraicGeometry.Scheme.IdealSheafData.ideal_comap_of_isOpenImmersion, AlgebraicGeometry.AffineSpace.SpecIso_inv_appTop_coord, AlgebraicGeometry.germ_stalkClosedPointIso_hom_assoc, toAlgHom_id, smoothSheafCommRing.forgetStalk_hom_comp_evalHom_assoc, AlgebraicGeometry.Scheme.evaluation_eq_zero_iff_notMem_basicOpen, AlgebraicGeometry.LocallyRingedSpace.restrictStalkIso_inv_eq_germ_apply, AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff_of_mem_mul, hom_comp, AlgebraicGeometry.LocallyRingedSpace.stalkMap_inv_hom_apply, AlgebraicGeometry.Scheme.stalkSpecializes_stalkMap_apply, AlgebraicGeometry.Scheme.zeroLocus_radical, HomTopology.isClosedEmbedding_hom, AlgebraicGeometry.StructureSheaf.globalSectionsIso_hom, AlgebraicGeometry.Scheme.Spec.residue_residueFieldIso_hom_assoc, AlgebraicGeometry.tilde.map_comp_assoc, AlgebraicGeometry.Scheme.IdealSheafData.map_ideal, hom_inv_apply, AlgebraicGeometry.Scheme.basicOpen_add_le, AlgebraicGeometry.Scheme.zeroLocus_map, AlgebraicGeometry.Scheme.Hom.finite_app, AlgebraicGeometry.Scheme.AffineZariskiSite.coequifibered_iff_forall_isLocalizationAway, AlgebraicGeometry.IsAffineOpen.fromSpec_preimage_zeroLocus, AlgebraicGeometry.Scheme.Modules.toOpen_fromTildeΓ_app_assoc, AlgebraicGeometry.FormallyUnramified.instIsSeparableCarrierResidueFieldCoeContinuousMapCarrierCarrierCommRingCatHomTopCatBaseOfLocallyOfFiniteType, AlgebraicGeometry.Scheme.Hom.ideal_ker_le, AlgebraicGeometry.AffineSpace.comp_homOfVector, AlgebraicGeometry.smooth_iff, AlgebraicGeometry.Scheme.zeroLocus_inf, AlgebraicGeometry.Scheme.IdealSheafData.ofIdeals_mono, AlgebraicGeometry.IsClosedImmersion.spec_of_quotient_mk, AlgebraicGeometry.germ_injective_of_isIntegral, ModuleCat.Derivation.desc_d, AlgebraicGeometry.isClosedImmersion_iff_isAffineHom, AlgebraicGeometry.AffineSpace.hom_ext_iff, forget₂CommSemiRing_preservesLimits, AlgebraicGeometry.ΓSpec.unop_locallyRingedSpaceAdjunction_counit_app', AlgebraicGeometry.Scheme.Hom.preimage_basicOpen, AlgebraicGeometry.stalkMap_toStalk, AlgebraicGeometry.Scheme.IdealSheafData.subschemeι_app, AlgEquiv.toUnder_inv_right_apply, HomTopology.mvPolynomialHomeomorph_symm_apply_hom, AlgebraicGeometry.Scheme.basicOpen_res_eq, AlgebraicGeometry.StructureSheaf.isLocalizedModule_toPushforwardStalkAlgHom, AlgebraicGeometry.Scheme.Hom.stalkFunctor_toImage_injective, AlgebraicGeometry.isNoetherian_iff_of_finite_iSup_eq_top, AlgebraicGeometry.Scheme.image_basicOpen, AlgebraicGeometry.tilde.isIso_toOpen_top, AlgebraicGeometry.LocallyRingedSpace.Hom.prop, TopCat.Presheaf.SubmonoidPresheaf.map, AlgebraicGeometry.SurjectiveOnStalks.surj_on_stalks, AlgebraicGeometry.Spec.fromSpecStalk_eq', AlgebraicGeometry.HasRingHomProperty.appTop, AlgebraicGeometry.LocallyRingedSpace.basicOpen_zero, commMon_forget₂_obj_coe, AlgebraicGeometry.isLocallyNoetherian_Spec, toAlgHom_apply, AlgebraicGeometry.Scheme.isNilpotent_iff_basicOpen_eq_bot, AlgebraicGeometry.SurjectiveOnStalks.stalkMap_surjective, AlgebraicGeometry.IsIntegral.component_integral, pushout_inl_tensorProdObjIsoPushoutObj_inv_right_assoc, HomTopology.isEmbedding_hom, AlgebraicGeometry.LocallyRingedSpace.restrictStalkIso_hom_eq_germ_apply, moduleCatExtendScalarsPseudofunctor_map, smoothSheafCommRing.ι_forgetStalk_inv_apply, AlgebraicGeometry.instIsFractionRingCarrierStalkCommRingCatPresheafFunctionField, AlgebraicGeometry.germ_stalkClosedPointIso_hom, AlgebraicGeometry.IsFinite.instHasAffinePropertyAndIsAffineFiniteCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTopHomAppTop, AlgebraicGeometry.SpecToEquivOfLocalRing_apply_fst, coproductCocone_pt, monoidAlgebra_map, AlgebraicGeometry.tilde.instIsLocalizedModuleCarrierCarrierOfCarrierStalkAbPresheafPrimeComplAsIdealHomToStalk, AlgebraicGeometry.Spec.map_apply, forgetToRingCat_obj, Colimits.cocone_naturality_components, AlgebraicGeometry.Spec.sheafedSpaceMap_hom_c_app, AlgebraicGeometry.Scheme.evaluation_naturality_apply, smoothSheafCommRing.eval_surjective, AlgebraicGeometry.Scheme.residue_surjective, AlgebraicGeometry.IsAffineOpen.iSup_basicOpen_eq_self_iff, AlgebraicGeometry.Scheme.basicOpen_one, AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff_of_mem, AlgebraicGeometry.targetAffineLocally_affineAnd_iff, smoothSheafCommRing.forgetStalk_hom_comp_evalHom, AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_stalk_isLocalHom, AlgebraicGeometry.Scheme.zeroLocus_eq_univ_iff_subset_nilradical_of_isCompact, FilteredColimits.forget_preservesFilteredColimits, AlgebraicGeometry.LocallyRingedSpace.Γevaluation_naturality_apply, moduleCatExtendScalarsPseudofunctor_obj, AlgebraicGeometry.StructureSheaf.toStalk_stalkSpecializes_apply, commAlgCatEquivUnder_counitIso, AlgebraicGeometry.Spec_zeroLocus_eq_zeroLocus, AlgebraicGeometry.Scheme.coprodPresheafObjIso_hom_snd, RingHom.surjective_iff_epi_and_finite, AlgebraicGeometry.Scheme.map_basicOpen, AlgebraicGeometry.Proj.zero_apply, PresheafOfModules.Derivation'.d_app, AlgebraicGeometry.Spec_carrier, TopCat.Sheaf.objSupIsoProdEqLocus_inv_snd, smoothSheafCommRing.ι_forgetStalk_inv_assoc, AlgebraicGeometry.IsAffineOpen.basicOpenSectionsToAffine_isIso, pushout_inl_tensorProdObjIsoPushoutObj_inv_right, AlgebraicGeometry.exists_smooth_of_formallySmooth_stalk, AlgebraicGeometry.Scheme.IdealSheafData.ideal_top, commAlgCatEquivUnder_inverse_obj_carrier, TopCat.Presheaf.instIsLocalizationCarrierObjOppositeOpensCarrierCommRingCatObjLocalizationPresheaf, AlgebraicGeometry.LocallyOfFiniteType.stalkMap, ModuleCat.Derivation.d_map, AlgebraicGeometry.Scheme.Hom.etale_appLE, AlgebraicGeometry.AffineSpace.map_SpecMap, AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalHom, ofHom_apply, AlgebraicGeometry.Scheme.IdealSheafData.isLocalization_away, RingHom.RespectsIso.cancel_left_isIso, AlgebraicGeometry.IsClosedImmersion.hasAffineProperty, AlgebraicGeometry.Scheme.Spec.algebraMap_residueFieldIso_inv, AlgHom.toUnder_right, AlgebraicGeometry.ProjectiveSpectrum.Proj.mk_mem_toSpec_base_apply, AlgebraicGeometry.AffineSpace.SpecIso_inv_over_assoc, AlgebraicGeometry.Scheme.eq_zeroLocus_of_isClosed_of_isAffine, AlgebraicGeometry.IsAffineOpen.comap_primeIdealOf_appLE, AlgebraicGeometry.Scheme.toSpecΓ_image_zeroLocus, PresheafOfModules.DifferentialsConstruction.relativeDifferentials'_map_d, AlgEquiv.toUnder_hom_right_apply, AlgebraicGeometry.IsAffineOpen.fromSpec_app_self_apply, AlgebraicGeometry.Scheme.stalkMap_hom_inv_apply, AlgebraicGeometry.Scheme.Hom.iInf_ker_openCover_map_comp_apply, HomTopology.continuous_apply, AlgebraicGeometry.eq_zero_of_basicOpen_eq_bot, AlgebraicGeometry.isLocalization_away_of_isAffine, AlgebraicGeometry.Scheme.basicOpen_appLE, AlgebraicGeometry.Scheme.Hom.stalkSpecializes_stalkMap_apply, AlgebraicGeometry.LocallyRingedSpace.isLocalHomStalkMap', forget_map_apply, AlgebraicGeometry.Scheme.IdealSheafData.ofIdealTop_ideal, commAlgCatEquivUnder_unitIso, AlgebraicGeometry.SurjectiveOnStalks.Spec_iff, AlgebraicGeometry.ΓSpecIso_hom_stalkClosedPointIso_inv, isLocalHom_of_isIso, AlgebraicGeometry.AffineSpace.toSpecMvPolyIntEquiv_apply, AlgebraicGeometry.Scheme.descResidueField_stalkClosedPointTo_fromSpecResidueField, forget₂Ring_preservesLimits, AlgebraicGeometry.IsAffineOpen.fromSpec_preimage_basicOpen, AlgebraicGeometry.Flat.flat_of_affine_subset, AlgebraicGeometry.IsClosedImmersion.Spec_iff, AlgebraicGeometry.Spec.algebraMap_stalkIso_inv, AlgebraicGeometry.LocallyRingedSpace.stalkMap_germ_apply, AlgebraicGeometry.stalkwise_Spec_map_iff, AlgebraicGeometry.HasAffineProperty.affineAnd_iff, AlgebraicGeometry.Scheme.default_asIdeal, AlgebraicGeometry.Scheme.IdealSheafData.ker_subschemeι_app, AlgebraicGeometry.Proj.add_apply, AlgebraicGeometry.Scheme.Hom.ι_toNormalization_assoc, equalizer_ι_isLocalHom', AlgebraicGeometry.Scheme.isLocalHom_stalkClosedPointTo', AlgebraicGeometry.Scheme.Hom.fromNormalization_app_assoc, FilteredColimits.forget₂Ring_preservesFilteredColimits, AlgebraicGeometry.Scheme.IdealSheafData.ideal_iInf, AlgebraicGeometry.stalkMap_toStalk_apply, AlgebraicGeometry.IsAffineOpen.primeIdealOf_eq_map_closedPoint, AlgebraicGeometry.Scheme.stalkMap_germ_apply, AlgebraicGeometry.Scheme.IdealSheafData.range_glueDataObjι, RingHom.surjective_of_epi_of_finite, AlgebraicGeometry.LocallyRingedSpace.instIsLocalRingCarrierStalkCommRingCatPresheaf, localization_unit_isIso, AlgebraicGeometry.Scheme.coprodPresheafObjIso_hom_fst, AlgebraicGeometry.Scheme.toOpen_eq, PresheafOfModules.DifferentialsConstruction.relativeDifferentials'_obj, TopCat.Presheaf.restrictOpenCommRingCat_apply, AlgebraicGeometry.Scheme.Hom.isIntegral_app, TopCat.Presheaf.SubmonoidPresheaf.toLocalizationPresheaf_app, AlgebraicGeometry.IsFinite.SpecMap_iff, PresheafOfModules.Derivation.d_app, AlgebraicGeometry.instIsOpenImmersionSigmaSpec, AlgebraicGeometry.instIsIsoModulesSpecOfCarrierFromTildeΓUnitOpensCarrierCarrierCommRingCatRingCatSheaf, AlgebraicGeometry.RingedSpace.basicOpen_mul, AlgebraicGeometry.Scheme.AffineZariskiSite.opensRange_relativeGluingData_map, coproductCoconeIsColimit_desc, AlgebraicGeometry.Flat.flat_appLE, AlgebraicGeometry.isIntegralHom_iff, AlgebraicGeometry.RingedSpace.basicOpen_res_eq, AlgebraicGeometry.Spec_zeroLocus, AlgebraicGeometry.LocallyOfFinitePresentation.finitePresentation_appLE, AlgebraicGeometry.Scheme.basicOpen_mul, ModuleCat.Derivation.d_add, AlgebraicGeometry.Scheme.IdealSheafData.coe_support_eq_eq_iInter_zeroLocus, PresheafOfModules.Monoidal.tensorHom_app, AlgebraicGeometry.IsPreimmersion.SpecMap_iff, PresheafOfModules.DifferentialsConstruction.relativeDifferentials'_map, AlgebraicGeometry.Scheme.Hom.finiteType_appLE, tensorProdIsoPushout_app, inv_hom_apply, AlgebraicGeometry.Scheme.zeroLocus_biInf_of_nonempty, AlgebraicGeometry.Flat.stalkMap, PresheafOfModules.Derivation'.app_apply, comp_apply, AlgebraicGeometry.isLocallyNoetherian_iff_of_iSup_eq_top, AlgebraicGeometry.StructureSheaf.instIsLocalizedModuleCarrierStalkAbPresheafOpensCarrierTopModuleStructurePresheafPrimeComplAsIdealToStalkₗ, AlgebraicGeometry.basicOpen_eq_of_affine', AlgebraicGeometry.AffineSpace.isoOfIsAffine_hom_appTop, AlgebraicGeometry.LocallyRingedSpace.stalkSpecializes_stalkMap_apply, AlgebraicGeometry.Spec.coe_toTop_map_hom_apply_asIdeal, smoothSheafCommRing.ι_forgetStalk_hom_apply, TopCat.Sheaf.objSupIsoProdEqLocus_hom_snd, AlgebraicGeometry.stalkToFiberRingHom_germ, AlgebraicGeometry.Scheme.IdealSheafData.supportSet_subset_zeroLocus, AlgebraicGeometry.instFaithfulModuleCatCarrierModulesSpecOfFunctor, TopCat.Sheaf.objSupIsoProdEqLocus_inv_eq_iff, AlgebraicGeometry.IsFinite.finite_app, AlgebraicGeometry.exists_eq_pow_mul_of_isAffineOpen, smoothSheafCommRing.ι_forgetStalk_hom_assoc, AlgebraicGeometry.finite_appTop_of_universallyClosed, AlgebraicGeometry.IsLocallyNoetherian.component_noetherian, AlgebraicGeometry.IsIntegralHom.integral_app, AlgebraicGeometry.tilde.map_add, commAlgCatEquivUnder_inverse_map, AlgebraicGeometry.isIso_SpecMap_stakMap_localization, coyonedaUnique_inv_app_hom_apply, AlgebraicGeometry.Scheme.Spec_fromSpecStalk', AlgebraicGeometry.FormallyUnramified.stalkMap, AlgebraicGeometry.specTargetImageFactorization_app_injective, AlgebraicGeometry.Scheme.IsGermInjectiveAt.cond, AlgebraicGeometry.Scheme.IdealSheafData.zeroLocus_inter_subset_supportSet, AlgebraicGeometry.ι_sigmaSpec_assoc, AlgebraicGeometry.Scheme.IdealSheafData.map_ideal', AlgebraicGeometry.HasRingHomProperty.Spec_iff, AlgebraicGeometry.IsOpenImmersion.app_ΓIso_hom_apply, AlgebraicGeometry.Scheme.IdealSheafData.ideal_sup, AlgebraicGeometry.LocallyRingedSpace.isLocalRing, AlgebraicGeometry.AffineSpace.isoOfIsAffine_hom, AlgebraicGeometry.StructureSheaf.stalkAlgebra_map, AlgebraicGeometry.tilde.toOpen_res, AlgebraicGeometry.tilde.toOpen_res_assoc, equalizer_ι_isLocalHom, AlgebraicGeometry.Scheme.stalkMap_inv_hom_apply, AlgebraicGeometry.affineAnd_apply, AlgebraicGeometry.Scheme.SpecMap_residue_apply, AlgebraicGeometry.tilde.isUnit_algebraMap_end_basicOpen, AlgebraicGeometry.HasRingHomProperty.iff_exists_appLE_locally, AlgebraicGeometry.targetAffineLocally_affineAnd_iff', AlgebraicGeometry.AffineSpace.reindex_appTop_coord, AlgebraicGeometry.Spec.algebraMap_stalkIso_inv_assoc, AlgebraicGeometry.smoothOfRelativeDimension_iff, AlgebraicGeometry.HasRingHomProperty.iff_exists_appLE, AlgebraicGeometry.Scheme.Hom.mem_smoothLocus, AlgebraicGeometry.Scheme.Hom.appLE_appIso_inv_apply, AlgebraicGeometry.Scheme.IdealSheafData.ideal_mono, AlgebraicGeometry.RingedSpace.isUnit_res_basicOpen, AlgebraicGeometry.IsAffineOpen.isoSpec_hom_apply, AlgebraicGeometry.ΓSpec.toSpecΓ_of, AlgebraicGeometry.Scheme.zeroLocus_span, AlgebraicGeometry.Scheme.IdealSheafData.ideal_iSup, AlgebraicGeometry.exists_appTop_π_eq_of_isLimit, AlgebraicGeometry.Proj.stalkIso'_germ, AlgebraicGeometry.IsAffineOpen.instAwayCarrierObjOppositeOpensCarrierCarrierCommRingCatSpecPresheafOpOpensBasicOpen, AlgebraicGeometry.formallyUnramified_iff, AlgebraicGeometry.Scheme.mem_basicOpen_top, moduleCatExtendScalarsPseudofunctor_mapComp, coyonedaUnique_hom_app_hom_apply, AlgebraicGeometry.Scheme.Opens.ι_image_basicOpen', AlgebraicGeometry.Spec_closedPoint, AlgebraicGeometry.IsIntegralHom.SpecMap_iff, AlgebraicGeometry.LocallyRingedSpace.preimage_basicOpen, AlgebraicGeometry.ProjectiveSpectrum.Proj.stalkMap_toSpec, AlgebraicGeometry.AffineSpace.homOverEquiv_apply, AlgebraicGeometry.Scheme.basicOpen_res, AlgebraicGeometry.StructureSheaf.toPushforwardStalk_comp, TopCat.Sheaf.objSupIsoProdEqLocus_inv_fst, AlgebraicGeometry.Smooth.exists_isStandardSmooth, AlgebraicGeometry.Scheme.Hom.stalkMap_hom_inv_apply, AlgebraicGeometry.Scheme.IdealSheafData.radical_ideal, HomTopology.mvPolynomialHomeomorph_apply_fst, AlgebraicGeometry.IsClosedImmersion.isAffine_surjective_of_isAffine, Under.equalizerFork_ι, AlgebraicGeometry.StructureSheaf.toPushforwardStalk_comp_assoc, AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.toFun_asIdeal, AlgebraicGeometry.IsAffineOpen.isLocalization_stalk, AlgebraicGeometry.Smooth.smooth_appLE, AlgebraicGeometry.AffineSpace.toSpecMvPolyIntEquiv_comp, AlgebraicGeometry.tilde.map_zero, AlgebraicGeometry.isField_of_universallyClosed, AlgebraicGeometry.IsAffineOpen.isoSpec_hom_base_apply, AlgebraicGeometry.Scheme.IdealSheafData.ideal_inf, AlgebraicGeometry.LocallyRingedSpace.Γevaluation_eq_zero_iff_notMem_basicOpen, AlgebraicGeometry.Scheme.Spec.algebraMap_residueFieldIso_inv_assoc, smoothSheafCommRing.ι_forgetStalk_hom, AlgebraicGeometry.ProjectiveSpectrum.Proj.awayToSection_germ, AlgebraicGeometry.Flat.iff_flat_stalkMap, AlgebraicGeometry.Scheme.IdealSheafData.subschemeι_app_surjective, AlgebraicGeometry.Scheme.preimage_zeroLocus, AlgebraicGeometry.Spec.map_base, AlgebraicGeometry.Scheme.IdealSheafData.coe_support_inter, commAlgCatEquivUnder_functor_map, AlgebraicGeometry.SpecToEquivOfLocalRing_symm_apply, piFan_pt, AlgebraicGeometry.Scheme.map_basicOpen_map, AlgebraicGeometry.IsIntegralHom.hasAffineProperty, AlgebraicGeometry.Scheme.germ_stalkClosedPointTo_Spec, AlgebraicGeometry.Scheme.basicOpen_zero, AlgebraicGeometry.IsAffineOpen.isLocalization_stalk', AlgebraicGeometry.Scheme.IdealSheafData.ideal_ofIdeals_le, monoidAlgebra_obj, AlgebraicGeometry.LocallyOfFinitePresentation.finitePresentation_of_affine_subset, AlgebraicGeometry.IsAffineOpen.fromSpec_image_basicOpen, AlgebraicGeometry.IsAffineOpen.basicOpen_union_eq_self_iff, AlgebraicGeometry.isFinite_iff, AlgebraicGeometry.etale_iff, AlgebraicGeometry.flat_and_surjective_SpecMap_iff, CategoryTheory.Iso.commRingCatIsoToRingEquiv_toRingHom, AlgebraicGeometry.Scheme.AffineZariskiSite.PreservesLocalization.opensRange_map, AlgebraicGeometry.StructureSheaf.comap_basicOpen, AlgebraicGeometry.tilde.functor_obj, AlgebraicGeometry.Scheme.mem_basicOpen'', AlgebraicGeometry.stalkClosedPointIso_inv, AlgebraicGeometry.specTargetImageRingHom_surjective, AlgebraicGeometry.tilde.toOpen_map_app_assoc, AlgebraicGeometry.Spec.topObj_forget, AlgebraicGeometry.basicOpen_eq_of_affine, hom_id, AlgebraicGeometry.IsAffineOpen.fromSpec_image_zeroLocus, isLocalHom_of_iso, AlgebraicGeometry.Scheme.IdealSheafData.mem_supportSet_iff_of_mem, AlgebraicGeometry.LocallyRingedSpace.isUnit_res_toΓSpecMapBasicOpen, AlgebraicGeometry.Scheme.affineBasisCover_map_range, AlgebraicGeometry.Scheme.IdealSheafData.supportSet_inter, AlgebraicGeometry.locallyOfFinitePresentation_iff, AlgebraicGeometry.AffineSpace.SpecIso_inv_over, commMon_forget₂_map, isLocalHom_comp, AlgebraicGeometry.LocallyRingedSpace.toΓSpec_preimage_zeroLocus_eq, AlgebraicGeometry.genericPoint_eq_bot_of_affine, AlgebraicGeometry.LocallyRingedSpace.SpecΓIdentity_hom_app, AlgebraicGeometry.tilde.map_comp, AlgebraicGeometry.basicOpen_eq_bot_iff, AlgebraicGeometry.Scheme.IdealSheafData.ker_glueDataObjι_appTop, AlgebraicGeometry.FormallyUnramified.formallyUnramified_appLE, AlgebraicGeometry.Spec.germ_stalkMapIso_hom, AlgebraicGeometry.Scheme.Hom.appIso_inv_app_apply, AlgebraicGeometry.HasRingHomProperty.stalkMap_of_respectsIso, localization_unit_isIso', AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv_over, AlgebraicGeometry.Spec.locallyRingedSpaceObj_presheaf, moduleCatExtendScalarsPseudofunctor_mapId, AlgebraicGeometry.Scheme.IdealSheafData.ideal_biInf, AlgebraicGeometry.Scheme.isLocalHom_stalkClosedPointTo, AlgebraicGeometry.IsAffineOpen.fromSpec_preimage_basicOpen', AlgebraicGeometry.Scheme.basicOpen_restrict, AlgebraicGeometry.IsAffineOpen.app_basicOpen_eq_away_map, forget_preservesLimits, AlgebraicGeometry.Scheme.IdealSheafData.glueDataObjι_ι, AlgebraicGeometry.Scheme.preimage_basicOpen, AlgebraicGeometry.SpecToEquivOfLocalRing_eq_iff, AlgebraicGeometry.Scheme.exists_le_and_germ_injective, AlgebraicGeometry.Scheme.Hom.flat_appLE, KaehlerDifferential.ext_iff, AlgebraicGeometry.AffineSpace.comp_homOfVector_assoc, forgetToRingCat_map_hom, AlgebraicGeometry.instIsIsoFunctorModuleCatCarrierUnitModulesSpecOfAdjunction, AlgebraicGeometry.tilde.map_neg, AlgebraicGeometry.LocallyRingedSpace.toStalk_stalkMap_toΓSpec, AlgebraicGeometry.Scheme.Hom.toImage_app, AlgebraicGeometry.exists_lift_of_germInjective_aux, AlgebraicGeometry.Scheme.Modules.Hom.app_smul, AlgebraicGeometry.LocallyRingedSpace.toΓSpecSheafedSpace_app_spec, AlgebraicGeometry.Scheme.Hom.stalkMap_inv_hom_apply, AlgebraicGeometry.Scheme.image_zeroLocus, AlgebraicGeometry.Scheme.basicOpen_eq_bot_iff_forall_evaluation_eq_zero, AlgebraicGeometry.HasRingHomProperty.iff_of_source_openCover, AlgebraicGeometry.LocallyRingedSpace.toΓSpecCBasicOpens_app, AlgebraicGeometry.LocallyRingedSpace.evaluation_eq_zero_iff_notMem_basicOpen, AlgebraicGeometry.Proj.res_apply, of_carrier, AlgebraicGeometry.LocallyRingedSpace.toΓSpecSheafedSpace_app_spec_assoc, AlgebraicGeometry.Scheme.isLocallyArtinianScheme_Spec, PresheafOfModules.germ_smul, AlgebraicGeometry.LocallyRingedSpace.toΓSpecCApp_spec, AlgebraicGeometry.AffineSpace.map_appTop_coord, AlgebraicGeometry.Scheme.toSpecΓ_preimage_basicOpen, AlgebraicGeometry.LocallyRingedSpace.isLocalHom_stalkMap_congr, prodFan_pt, AlgebraicGeometry.isIso_fromTildeΓ_iff, AlgebraicGeometry.Scheme.Hom.normalizationObjIso_hom_val, AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.num_mem_carrier_iff, AlgebraicGeometry.LocallyRingedSpace.toΓSpecSheafedSpace_hom_c_app, moduleCatRestrictScalarsPseudofunctor_obj, AlgebraicGeometry.tilde.map_id_assoc, AlgebraicGeometry.Scheme.isArtinianScheme_Spec, smoothSheafCommRing.forgetStalk_hom_comp_evalHom_apply, AlgebraicGeometry.isField_of_isIntegral_of_subsingleton, AlgebraicGeometry.instIsNoetherianRingCarrierStalkCommRingCatPresheafOfIsLocallyNoetherian, AlgebraicGeometry.Spec_sheaf, AlgebraicGeometry.isReduced_stalk_of_isReduced, AlgebraicGeometry.Scheme.Hom.instIsLocalHomCarrierStalkCommRingCatPresheafCoeContinuousMapCarrierCarrierHomTopCatBaseRingHomHomStalkMap, TopCat.Presheaf.submonoidPresheafOfStalk_obj, AlgebraicGeometry.StructureSheaf.algebraMap_pushforward_stalk, AlgebraicGeometry.isLocalization_basicOpen_of_qcqs, AlgebraicGeometry.Scheme.IdealSheafData.strictMono_ideal, AlgebraicGeometry.LocallyRingedSpace.stalkMap_hom_inv_apply, AlgebraicGeometry.Scheme.stalkClosedPointTo_comp, AlgebraicGeometry.LocallyRingedSpace.evaluation_naturality_apply, AlgebraicGeometry.map_injective_of_isIntegral, AlgebraicGeometry.exists_eq_pow_mul_of_isCompact_of_isQuasiSeparated, AlgebraicGeometry.AffineSpace.homOfVector_appTop_coord, AlgebraicGeometry.Scheme.Spec_map_residue_apply, AlgebraicGeometry.sourceAffineLocally_morphismRestrict, AlgebraicGeometry.Proj.stalkIso'_symm_mk, CommAlgCat.forget₂_commRingCat_map, AlgebraicGeometry.Scheme.isoSpec_image_zeroLocus, RingHom.RespectsIso.arrow_mk_iso_iff, AlgebraicGeometry.Scheme.IdealSheafData.ideal_map_of_isAffineHom, AlgebraicGeometry.Scheme.Spec_stalkClosedPointTo_fromSpecStalk, AlgebraicGeometry.LocallyRingedSpace.isLocalHomValStalkMap, AlgebraicGeometry.localRingHom_comp_stalkIso, AlgebraicGeometry.Spec.toTop_obj_carrier, AlgebraicGeometry.Scheme.localRingHom_comp_stalkIso, AlgebraicGeometry.RingedSpace.mem_basicOpen, AlgebraicGeometry.Scheme.Hom.app_injective, AlgebraicGeometry.Scheme.Opens.toSpecΓ_preimage_zeroLocus, AlgebraicGeometry.LocallyRingedSpace.toΓSpec_preimage_basicOpen_eq, AlgebraicGeometry.StructureSheaf.algebraMap_germ_apply, AlgebraicGeometry.Scheme.Spec.residue_residueFieldIso_hom, AlgebraicGeometry.Scheme.AffineZariskiSite.mem_grothendieckTopology_iff_sectionsOfPresieve, AlgebraicGeometry.Scheme.IdealSheafData.map_ideal_basicOpen, coproductCocone_ι, AlgebraicGeometry.Scheme.localRingHom_comp_stalkIso_apply, AlgebraicGeometry.instIsIsoModulesSpecOfCarrierFromTildeΓFreeOpensCarrierCarrierCommRingCat, AlgebraicGeometry.Scheme.Hom.isOpen_smoothLocus, AlgebraicGeometry.Scheme.Hom.fromNormalization_app, AlgebraicGeometry.Spec_stalkClosedPointIso, AlgebraicGeometry.flat_iff, AlgebraicGeometry.Scheme.zeroLocus_map_of_eq, AlgebraicGeometry.HasRingHomProperty.iff_of_isAffine, instIsRightAdjointForgetRingHomCarrier, AlgebraicGeometry.Scheme.IdealSheafData.le_def, AlgebraicGeometry.ΓSpec.toSpecΓ_unop, AlgebraicGeometry.Spec.locallyRingedSpaceObj_sheaf, Under.equalizer_comp, AlgebraicGeometry.tilde.instIsLocalizedModuleCarrierCarrierObjOppositeOpensCarrierCarrierCommRingCatSpecModuleCatPresheafModulesSheafModulesSpecToSheafOpBasicOpenPowersHomToOpen, AlgebraicGeometry.Scheme.isNilpotent_iff_basicOpen_eq_bot_of_isCompact, AlgebraicGeometry.affine_isReduced_iff, Under.equalizerFork'_ι, AlgebraicGeometry.Scheme.IdealSheafData.supportSet_eq_iInter_zeroLocus, AlgebraicGeometry.IsAffineOpen.appLE_eq_away_map, AlgebraicGeometry.Scheme.Modules.map_smul, AlgebraicGeometry.RingedSpace.mem_basicOpen', AlgebraicGeometry.Scheme.mem_basicOpen, AlgebraicGeometry.Scheme.Spec_stalkClosedPointTo_fromSpecStalk_assoc, AlgebraicGeometry.isNoetherian_Spec, AlgebraicGeometry.IsOpenImmersion.map_ΓIso_inv_apply, AlgebraicGeometry.LocallyRingedSpace.toΓSpecCApp_iff, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.ofRestrict_invApp_apply, AlgebraicGeometry.IsAffineOpen.Spec_basicOpen, AlgebraicGeometry.Spec.sheafedSpaceObj_presheaf, AlgebraicGeometry.Spec.map_app, AlgebraicGeometry.homogeneousLocalizationToStalk_stalkToFiberRingHom, AlgebraicGeometry.IsAffineOpen.ideal_ext_iff, AlgebraicGeometry.Spec.map_base_apply, AlgebraicGeometry.Scheme.Hom.stalkMap_surjective, AlgebraicGeometry.tilde.toOpen_map_app, pushout_inr_tensorProdObjIsoPushoutObj_inv_right, AlgebraicGeometry.IsAffineOpen.mem_ideal_iff, AlgebraicGeometry.Scheme.toSpecΓ_base, AlgebraicGeometry.Smooth.iff_forall_exists_isStandardSmooth
forget_obj_eq_coe 📖	MathDef	—
fullyFaithfulForget₂ToRingCat 📖	CompOp	—
hasForgetToAddCommMonCat 📖	CompOp	2 math math: forget₂CommSemiRing_preservesLimitsOfSize, forget₂CommSemiRing_preservesLimits
hasForgetToRingCat 📖	CompOp	23 math math: PresheafOfModules.Monoidal.tensorObj_obj, forget₂Ring_preservesLimitsOfSize, AlgebraicGeometry.StructureSheaf.instIsScalarTowerCarrierStalkCommRingCatStructurePresheafInCommRingCatCarrierAbPresheafOpensCarrierTopModuleStructurePresheaf, instFullRingCatForget₂RingHomCarrierCarrier, PresheafOfModules.Derivation.d_map, PresheafOfModules.Derivation.postcomp_d_apply, PresheafOfModules.Derivation.d_one, PresheafOfModules.Derivation.d_mul, PresheafOfModules.Monoidal.tensorObj_map_tmul, PresheafOfModules.Derivation.congr_d, forgetToRingCat_obj, PresheafOfModules.Derivation'.d_app, PresheafOfModules.DifferentialsConstruction.relativeDifferentials'_map_d, forget₂Ring_preservesLimits, FilteredColimits.forget₂Ring_preservesFilteredColimits, PresheafOfModules.DifferentialsConstruction.relativeDifferentials'_obj, PresheafOfModules.Derivation.d_app, PresheafOfModules.Monoidal.tensorHom_app, PresheafOfModules.DifferentialsConstruction.relativeDifferentials'_map, PresheafOfModules.Derivation'.app_apply, AlgebraicGeometry.StructureSheaf.instIsLocalizedModuleCarrierStalkAbPresheafOpensCarrierTopModuleStructurePresheafPrimeComplAsIdealToStalkₗ, forgetToRingCat_map_hom, PresheafOfModules.germ_smul
instCategory 📖	CompOp	1633 math math: tensorProd_map_right, AlgebraicGeometry.Scheme.toSpecΓ_apply, PresheafOfModules.Monoidal.tensorObj_obj, AlgebraicGeometry.Scheme.Opens.ι_image_basicOpen, AlgebraicGeometry.Γ_map_morphismRestrict, AlgebraicGeometry.Scheme.Hom.smoothLocus_eq_top_iff, AlgebraicGeometry.Scheme.Modules.pushforward_obj_presheaf_map, smoothSheafCommRing.ι_forgetStalk_inv, AlgebraicGeometry.Scheme.ΓSpecIso_inv_naturality_assoc, AlgebraicGeometry.Proj.awayMap_awayToSection_assoc, AlgebraicGeometry.QuasiCompact.isCompact_preimage_singleton, AlgebraicGeometry.Scheme.fromSpecStalk_closedPoint, AlgebraicGeometry.Scheme.coprodPresheafObjIso_hom_fst_assoc, AlgebraicGeometry.Scheme.Hom.toPartialMap_hom, AlgebraicGeometry.irreducibleSpace_of_isIntegral, AlgebraicGeometry.Scheme.height_of_isClosed, AlgebraicGeometry.Scheme.map_PrimeSpectrum_basicOpen_of_affine, forget₂Ring_preservesLimitsOfSize, AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec_injective, AlgebraicGeometry.Scheme.isoOfEq_inv, AlgebraicGeometry.StructureSheaf.instIsScalarTowerCarrierStalkCommRingCatStructurePresheafInCommRingCatCarrierAbPresheafOpensCarrierTopModuleStructurePresheaf, AlgebraicGeometry.LocallyRingedSpace.stalkMap_hom_inv_assoc, AlgebraicGeometry.Spec.toPresheafedSpace_obj, AlgebraicGeometry.IsAffineOpen.isoSpec_inv_appTop, AlgebraicGeometry.Spec.map_eq_id, AlgebraicGeometry.Scheme.zeroLocus_iInf, AlgebraicGeometry.Scheme.Modules.pushforwardId_inv_app_app, HomTopology.mvPolynomialHomeomorph_apply_snd, AlgebraicGeometry.IsAffineOpen.map_fromSpec_assoc, AlgebraicGeometry.IsAffineOpen.isLocalization_of_eq_basicOpen, AlgebraicGeometry.Scheme.Hom.opensRange_pullbackSnd, AlgebraicGeometry.LocallyRingedSpace.residueFieldMap_comp, AlgebraicGeometry.Scheme.AffineZariskiSite.instFaithfulOpensToOpensFunctor, AlgebraicGeometry.LocallyRingedSpace.instFaithfulSheafedSpaceCommRingCatForgetToSheafedSpace, RingHom.isStableUnderCobaseChange_toMorphismProperty_iff, AlgebraicGeometry.instCompactSpaceCarrierCarrierCommRingCatEqualizerSchemeOfQuasiSeparatedSpace, AlgebraicGeometry.Scheme.Spec_map_stalkMap_fromSpecStalk, AlgebraicGeometry.Scheme.Hom.image_preimage_eq_opensRange_inter, AlgebraicGeometry.ProjectiveSpectrum.Proj.awayToSection_apply, AlgebraicGeometry.Scheme.Cover.LocallyDirected.directed, instFullRingCatForget₂RingHomCarrierCarrier, AlgebraicGeometry.Scheme.Modules.instFaithfulPresheafOfModulesToPresheafOfModules, instNontrivialCarrierStalkCommRingCatPresheafSmoothSheafCommRing, AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction_counit_app, AlgebraicGeometry.GeometricallyIrreducible.geometrically_irreducibleSpace, AlgebraicGeometry.LocallyRingedSpace.Γ_def, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.instIsOpenImmersionCommRingCatMapSheafedSpaceForgetToSheafedSpace, AlgebraicGeometry.Scheme.IsQuasiAffine.toIsImmersion, AlgebraicGeometry.StructureSheaf.globalSectionsIso_inv, forget_preservesLimitsOfSize, AlgebraicGeometry.iSup_affineOpens_eq_top, AlgebraicGeometry.isAffineOpen_top, forgetReflectIsos, AlgebraicGeometry.RingedSpace.basicOpen_res, AlgebraicGeometry.Scheme.Hom.denseRange, AlgebraicGeometry.LocallyRingedSpace.restrict_presheaf_obj, AlgebraicGeometry.Scheme.isEmpty_pullback_iff, AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv, AlgebraicGeometry.Scheme.Hom.quasiFiniteLocus_comp, AlgebraicGeometry.Scheme.Pullback.Triplet.tensorCongr_trans_hom, AlgebraicGeometry.Scheme.app_eq, AlgebraicGeometry.Scheme.Hom.closePoints_subset_preimage_closedPoints, AlgebraicGeometry.Scheme.support_nilradical, AlgebraicGeometry.opensCone_pt, Localization.epi', AlgebraicGeometry.Scheme.zeroLocus_empty_eq_univ, AlgebraicGeometry.coprodSpec_apply, AlgebraicGeometry.IsAffineOpen.isCompact, AlgebraicGeometry.morphismRestrict_base_coe, AlgebraicGeometry.Scheme.Hom.fiberHomeo_apply, AlgebraicGeometry.surjectiveOnStalks_iff, AlgebraicGeometry.StructureSheaf.comap_id, AlgebraicGeometry.instCompactSpaceCarrierCarrierCommRingCatPullbackSchemeOfQuasiCompact, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.forgetToPresheafedSpace_preservesPullback_of_right, pushoutCocone_pt, AlgebraicGeometry.Scheme.Hom.instIsContinuousOpensOpensFunctorGrothendieckTopologyCarrierCarrierCommRingCat, AlgebraicGeometry.instIsDomainCarrierStalkCommRingCatPresheafOfIsIntegral, AlgebraicGeometry.instIsIsoSchemeCoprodComparisonOppositeCommRingCatSpec, AlgebraicGeometry.IsAffineOpen.map_fromSpec, AlgebraicGeometry.ΓSpec.isIso_adjunction_counit, AlgebraicGeometry.Spec_Γ_naturality, smoothSheafCommRing.nonunits_stalk, AlgebraicGeometry.Scheme.affineBasisCover_is_basis, AlgebraicGeometry.AffineScheme.Spec_faithful, AlgebraicGeometry.IsLocallyArtinian.isArtinianRing_presheaf_obj, AlgebraicGeometry.Scheme.Opens.toSpecΓ_naturality, AlgebraicGeometry.Scheme.Opens.toSpecΓ_SpecMap_presheaf_map_top, AlgebraicGeometry.range_eq_range_of_surjective, AlgebraicGeometry.LocallyRingedSpace.notMem_prime_iff_unit_in_stalk, AlgebraicGeometry.Scheme.restrictFunctorΓ_inv_app, AlgebraicGeometry.Scheme.Cover.isOpenMap_fromGlued, AlgebraicGeometry.Scheme.Modules.restrictAdjunction_counit_app_app, AlgebraicGeometry.ProjectiveSpectrum.Proj.toStalk_specStalkEquiv, AlgebraicGeometry.instGeometricallyIrreducibleMorphismRestrict, AlgebraicGeometry.preservesColimit_Γ, AlgebraicGeometry.instIsDomainCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTopOfIsIntegral, AlgebraicGeometry.Scheme.Pullback.Triplet.ofPoint_x, AlgebraicGeometry.StructureSheaf.comap_comp, AlgebraicGeometry.Scheme.Opens.mem_ι_image_iff, AlgebraicGeometry.LocallyRingedSpace.restrictStalkIso_hom_eq_germ, AlgebraicGeometry.Scheme.Hom.exists_isIso_morphismRestrict_toNormalization, AlgebraicGeometry.AffineSpace.toSpecMvPolyIntEquiv_symm_apply, AlgebraicGeometry.morphismRestrict_ι_assoc, AlgebraicGeometry.functionField_isFractionRing_of_isAffineOpen, AlgebraicGeometry.Scheme.IdealSheafData.ideal_sSup, AlgebraicGeometry.StructureSheaf.IsLocalization.to_stalk, AlgebraicGeometry.Scheme.Hom.stalkMap_congr_point, AlgebraicGeometry.ValuativeCriterion.Existence.specializingMap, AlgebraicGeometry.Scheme.fromSpecStalk_toSpecΓ_assoc, AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec_fromSpec, AlgebraicGeometry.Scheme.Modules.pushforwardCongr_hom_app_app, commRingCat_hasStrictTerminalObjects, AlgebraicGeometry.AffineSpace.SpecIso_hom_appTop, AlgebraicGeometry.LocallyRingedSpace.basicOpen_eq_bot_iff_forall_evaluation_eq_zero, AlgebraicGeometry.Scheme.Modules.pushforward_obj_obj, AlgebraicGeometry.Scheme.Hom.range_fiberι, AlgebraicGeometry.Spec.locallyRingedSpaceObj_presheaf_map, AlgebraicGeometry.Scheme.Hom.preimage_bot, AlgebraicGeometry.LocallyRingedSpace.restrictStalkIso_inv_eq_germ_assoc, AlgebraicGeometry.Scheme.Cover.iUnion_range, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.inv_naturality, AlgebraicGeometry.Scheme.mem_basicOpen', AlgebraicGeometry.instPrespectralSpaceCarrierCarrierCommRingCat, AlgebraicGeometry.pullbackRestrictIsoRestrict_hom_morphismRestrict_assoc, AlgebraicGeometry.Spec.map_surjective, AlgebraicGeometry.Scheme.Hom.mem_opensRange, AlgebraicGeometry.Scheme.zeroLocus_eq_univ_iff_subset_nilradical, AlgebraicGeometry.Scheme.Opens.topIso_inv, AlgebraicGeometry.Scheme.SpecToEquivOfField_eq_iff, AlgebraicGeometry.ProjectiveSpectrum.Proj.toSpec_preimage_basicOpen, AlgebraicGeometry.Proj.pow_apply, AlgebraicGeometry.isCompact_iff_exists, id_apply, AlgebraicGeometry.Scheme.Modules.instPreservesLimitsPresheafAbCarrierCommRingCatToPresheaf, AlgebraicGeometry.Spec_presheaf, AlgebraicGeometry.Scheme.RationalMap.dense_domain, AlgebraicGeometry.HasRingHomProperty.stalkMap, AlgebraicGeometry.Scheme.Hom.resLE_eq_morphismRestrict, AlgebraicGeometry.LocallyRingedSpace.homMk_toHom, AlgebraicGeometry.StructureSheaf.toOpen_comp_comap_apply, AlgebraicGeometry.Scheme.Hom.iUnion_support_ker_openCover_map_comp, AlgebraicGeometry.Scheme.inv_app, AlgebraicGeometry.LocallyRingedSpace.id_toShHom', AlgebraicGeometry.Scheme.Hom.id_appTop, AlgebraicGeometry.Proj.basicOpenIsoAway_hom, smoothSheafCommRing.ι_evalHom_apply, HomTopology.instCompactSpaceHomOfIsTopologicalRingOfT1SpaceOfCarrier, AlgebraicGeometry.image_morphismRestrict_preimage, AlgebraicGeometry.Scheme.IdealSheafData.gc, AlgebraicGeometry.Scheme.emptyTo_base, AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec_hom_apply_asIdeal, AlgebraicGeometry.Scheme.zeroLocus_mul, AlgebraicGeometry.Spec.map_injective, AlgebraicGeometry.Proj.mul_apply, AlgebraicGeometry.StructureSheaf.comap_id', AlgebraicGeometry.Scheme.OpenCover.finiteSubcover_f, AlgebraicGeometry.SpecMap_preimage_basicOpen, mkUnder_ext_iff, free_obj_coe, AlgebraicGeometry.Scheme.ideal_ker_le_ker_ΓSpecIso_inv_comp, AlgebraicGeometry.Scheme.stalkMap_congr_assoc, smoothSheafCommRing.eval_germ, AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec_surjective, AlgebraicGeometry.instIsGermInjectiveAtCoeContinuousMapCarrierCarrierCommRingCatHomTopCatBaseOfIsOpenImmersion, AlgebraicGeometry.Scheme.isoSpec_inv_toSpecΓ, AlgebraicGeometry.Scheme.Pullback.SpecTensorTo_SpecOfPoint, AlgebraicGeometry.Scheme.hom_inv_apply, moduleCatRestrictScalarsPseudofunctor_mapComp, AlgebraicGeometry.Scheme.Hom.range_asFiberHom, AlgebraicGeometry.isIso_ΓSpec_adjunction_unit_app_basicOpen, forget₂CommSemiRing_preservesLimitsOfSize, AlgebraicGeometry.isNoetherian_iff_of_finite_affine_openCover, AlgebraicGeometry.Scheme.Hom.tendsto_cofinite_cofinite, AlgebraicGeometry.Scheme.ι_toIso_inv, AlgebraicGeometry.localRingHom_comp_stalkIso_apply, CommAlgCat.forget₂_commRingCat_obj, AlgebraicGeometry.isPullback_opens_inf, AlgebraicGeometry.SmoothOfRelativeDimension.exists_isStandardSmoothOfRelativeDimension, AlgebraicGeometry.Spec.sheafedSpaceObj_carrier, AlgebraicGeometry.noetherianSpace_of_isAffine, AlgebraicGeometry.Scheme.Hom.preimageIso_hom_ι_assoc, forget_obj, AlgebraicGeometry.IsClosedImmersion.base_closed, AlgebraicGeometry.quasiCompact_iff, PresheafOfModules.Derivation.d_map, AlgebraicGeometry.Scheme.IdealSheafData.equivOfIsAffine_apply, AlgebraicGeometry.ValuativeCriterion.Existence.instIsLocalHomCarrierRingHomHomHomCommRingCat, free_map_coe, AlgebraicGeometry.StructureSheaf.instAtPrimeCarrierStalkCommRingCatStructurePresheafInCommRingCatAsIdeal, AlgebraicGeometry.StructureSheaf.toStalk_stalkSpecializes_assoc, AlgebraicGeometry.isEmpty_of_commSq_sigmaι_of_ne, AlgebraicGeometry.Scheme.Modules.Hom.zero_app, AlgebraicGeometry.AffineScheme.instIsEquivalenceOppositeCommRingCatOpRightOpΓ, toAlgHom_comp, AlgebraicGeometry.Scheme.IdealSheafData.equivOfIsAffine_symm_apply, AlgebraicGeometry.Scheme.congr_app, AlgebraicGeometry.Scheme.IdealSheafData.ideal_mul, AlgebraicGeometry.IsZariskiLocalAtTarget.restrict, AlgebraicGeometry.Scheme.Opens.stalkIso_inv, AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.structurePresheafInCommRing_obj_carrier, AlgebraicGeometry.Scheme.SpecΓIdentity_hom_app, smoothSheafCommRing.instLocalRing_stalk, AlgebraicGeometry.locallyQuasiFinite_iff, AlgebraicGeometry.Scheme.preimage_comp, AlgebraicGeometry.Scheme.Hom.stalkMap_congr_hom_assoc, AlgebraicGeometry.Scheme.Cover.RelativeGluingData.toBase_preimage_eq_opensRange_ι, AlgebraicGeometry.Scheme.IdealSheafData.support_mul, AlgebraicGeometry.sourceLocalClosure.iff_forall_exists, AlgebraicGeometry.IsFinite.instMorphismRestrict, AlgebraicGeometry.StructureSheaf.toOpen_comp_comap, AlgebraicGeometry.AffineScheme.instIsEquivalenceOppositeCommRingCatRightOpΓ, AlgebraicGeometry.Scheme.Pullback.Triplet.isPullback_SpecMap_tensor, AlgebraicGeometry.Scheme.forget_map, AlgebraicGeometry.sigmaMk_mk, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toLocallyRingedSpace_toSheafedSpace, AlgebraicGeometry.affineLocally_iff_forall_isAffineOpen, AlgebraicGeometry.Scheme.Pullback.Triplet.specTensorTo_base_snd, AlgebraicGeometry.LocallyRingedSpace.stalkMap_congr_point_assoc, AlgebraicGeometry.Spec.locallyRingedSpaceMap_id, AlgebraicGeometry.Scheme.isoSpec_inv_preimage_zeroLocus, AlgebraicGeometry.LocallyRingedSpace.preservesColimits_forgetToSheafedSpace, coyoneda_map_app, AlgebraicGeometry.ProjectiveSpectrum.Proj.toOpen_toSpec_val_c_app_assoc, AlgebraicGeometry.IsPreimmersion.isEmbedding, AlgebraicGeometry.Spec.locallyRingedSpaceObj_presheaf', AlgebraicGeometry.functionField_isScalarTower, AlgebraicGeometry.Scheme.Modules.toOpen_fromTildeΓ_app, AlgebraicGeometry.Scheme.Pullback.tensorCongr_SpecTensorTo, AlgebraicGeometry.Scheme.Hom.SpecMap_residueFieldMap_fromSpecResidueField_assoc, ofHom_id, AlgebraicGeometry.Scheme.exists_germ_injective, AlgebraicGeometry.Scheme.forgetToTop_map, AlgebraicGeometry.IsOpenImmersion.iff_isIso_stalkMap, AlgebraicGeometry.Scheme.Cover.instIsIsoCommRingCatStalkMapFromGlued, AlgebraicGeometry.LocallyRingedSpace.toΓSpecSheafedSpace_app_eq, AlgebraicGeometry.Scheme.Hom.preimage_top, AlgebraicGeometry.Scheme.coprodPresheafObjIso_hom_snd_assoc, AlgebraicGeometry.Scheme.Hom.toNormalization_app_preimage, AlgebraicGeometry.Scheme.Pullback.carrierEquiv_eq_iff, AlgebraicGeometry.Proj.one_apply, AlgebraicGeometry.AffineScheme.mk_obj, AlgebraicGeometry.Scheme.Hom.toImage_app_injective, AlgebraicGeometry.Scheme.isoSpec_hom, AlgebraicGeometry.instHasAffinePropertyQuasiSeparatedQuasiSeparatedSpaceCarrierCarrierCommRingCat, AlgebraicGeometry.Scheme.ι_toIso_inv_assoc, AlgebraicGeometry.Scheme.Hom.id_preimage, AlgebraicGeometry.Flat.instMorphismRestrict, AlgebraicGeometry.StructureSheaf.to_basicOpen_epi, AlgebraicGeometry.IsAffineOpen.opensRange_fromSpec, AlgebraicGeometry.Scheme.IdealSheafData.ideal_le_ker_glueDataObjι, AlgebraicGeometry.SpecToEquivOfLocalRing_apply_snd_coe, AlgebraicGeometry.instQuasiCompactMorphismRestrict, AlgebraicGeometry.Scheme.forget_obj, AlgebraicGeometry.Scheme.eqToHom_c_app, AlgebraicGeometry.IsAffineOpen.self_le_iSup_basicOpen_iff, AlgebraicGeometry.Scheme.zeroLocus_biInf, AlgebraicGeometry.Proj.basicOpenToSpec_app_top, AlgebraicGeometry.Scheme.Hom.image_preimage_le, AlgebraicGeometry.Scheme.AffineZariskiSite.generate_presieveOfSections, AlgebraicGeometry.Scheme.Hom.image_top_eq_opensRange, AlgebraicGeometry.Scheme.forgetToTop_obj, pushoutCocone_inl, AlgebraicGeometry.IsOpenImmersion.isOpen_range, AlgebraicGeometry.Scheme.affineOpenCover_idx, AlgebraicGeometry.Scheme.Hom.fromNormalization_preimage, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.stalk_iso, AlgebraicGeometry.Scheme.toSpecΓ_naturality_assoc, AlgebraicGeometry.LocallyRingedSpace.iso_inv_base_hom_base_apply, AlgebraicGeometry.Scheme.Pullback.Triplet.snd_SpecTensorTo_apply, AlgebraicGeometry.Scheme.Modules.pushforwardComp_inv_app_app, coyoneda_obj_obj_carrier, AlgebraicGeometry.SurjectiveOnStalks.iff_of_isAffine, isFinitelyPresentable_hom, AlgebraicGeometry.SpecMap_ΓSpecIso_hom, AlgebraicGeometry.Scheme.compactSpace_of_isAffine, AlgebraicGeometry.IsAffineOpen.isLocalization_basicOpen, AlgebraicGeometry.Scheme.zeroLocus_iUnion, hasLimit, AlgebraicGeometry.IsReduced.component_reduced, coyoneda_obj_map, AlgebraicGeometry.IsIntegralHom.isIntegral_app, AlgebraicGeometry.Scheme.ker_ideal_of_isPullback_of_isOpenImmersion, PresheafOfModules.Derivation.postcomp_d_apply, AlgebraicGeometry.Scheme.presieve₀_mem_precoverage_iff, AlgebraicGeometry.Scheme.comp_base, AlgebraicGeometry.LocallyRingedSpace.Γ_obj_op, AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv_appTop_coord, PresheafOfModules.Derivation.d_one, AlgebraicGeometry.Scheme.Hom.image_preimage_eq_opensRange_inf, AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv_over_assoc, AlgebraicGeometry.ProjectiveSpectrum.Proj.awayToΓ_ΓToStalk, AlgebraicGeometry.isIso_iff_isIso_stalkMap, AlgebraicGeometry.LocallyRingedSpace.stalkMap_inv_hom, AlgebraicGeometry.StructureSheaf.comap_const, AlgebraicGeometry.Scheme.Hom.image_iSup₂, AlgebraicGeometry.Scheme.isoSpec_hom_naturality, AlgebraicGeometry.Scheme.zeroLocus_singleton, AlgebraicGeometry.instIsEmptyCarrierCarrierCommRingCatEmptyCollectionScheme, AlgebraicGeometry.Scheme.toSpecΓ_isoSpec_inv_assoc, AlgebraicGeometry.LocallyRingedSpace.component_nontrivial, AlgebraicGeometry.Scheme.Pullback.residueFieldCongr_inv_residueFieldMap_ofPoint, AlgebraicGeometry.Scheme.IdealSheafData.ideal_map, HomTopology.continuous_precomp, AlgebraicGeometry.IsOpenImmersion.range_pullbackSnd, AlgebraicGeometry.Proj.sub_apply, AlgebraicGeometry.IsImmersion.instMorphismRestrict, AlgebraicGeometry.Scheme.Hom.isDiscrete_preimage_singleton, AlgebraicGeometry.Scheme.Hom.comp_app, AlgebraicGeometry.Scheme.IdealSheafData.range_glueDataObjι_ι_eq_support_inter, PresheafOfModules.Derivation.d_mul, AlgebraicGeometry.Scheme.hom_base_inv_base, AlgebraicGeometry.LocallyRingedSpace.stalkMap_congr, AlgebraicGeometry.isNoetherian_iff, Under.preservesFiniteLimits_of_flat, AlgebraicGeometry.StructureSheaf.toStalk_stalkSpecializes, AlgebraicGeometry.IsOpenImmersion.range_pullback_fst_of_right, AlgebraicGeometry.Scheme.toSpecΓ_naturality, AlgebraicGeometry.Scheme.Hom.quasiFiniteLocus_eq_top, AlgebraicGeometry.pointEquivClosedPoint_apply_coe, AlgebraicGeometry.IsNoetherian.toCompactSpace, pushout_inr_tensorProdObjIsoPushoutObj_inv_right_assoc, AlgebraicGeometry.IsIntegral.nonempty, AlgebraicGeometry.Scheme.Hom.ι_toNormalization, AlgebraicGeometry.LocallyRingedSpace.Γ_Spec_left_triangle, AlgebraicGeometry.StructureSheaf.comap_apply, AlgebraicGeometry.Scheme.GlueData.instIsOpenImmersionCommRingCatFSheafedSpaceToSheafedSpaceGlueDataToLocallyRingedSpaceGlueData, AlgebraicGeometry.Scheme.IdealSheafData.ideal_pow, AlgebraicGeometry.RingedSpace.mem_top_basicOpen, AlgebraicGeometry.Spec.faithful, AlgebraicGeometry.Spec.preimage_id, AlgebraicGeometry.IsNoetherian.noetherianSpace, AlgebraicGeometry.Scheme.range_fromSpecStalk, AlgebraicGeometry.Scheme.homeoOfIso_symm, AlgebraicGeometry.Scheme.comp_coeBase_assoc, AlgebraicGeometry.instCompactSpaceCarrierCarrierCommRingCatPullbackSchemeOfQuasiCompact_1, AlgebraicGeometry.IsAffineOpen.ideal_le_iff, hasLimitsOfShape, AlgebraicGeometry.stalkToFiberRingHom_homogeneousLocalizationToStalk, AlgebraicGeometry.quasiCompact_affineProperty_iff_quasiSeparatedSpace, AlgebraicGeometry.Scheme.Hom.congr_app, AlgebraicGeometry.instFullOppositeCommRingCatLocallyRingedSpaceToLocallyRingedSpace, AlgebraicGeometry.Scheme.Hom.preimage_iSup, AlgebraicGeometry.Scheme.Hom.toPartialMap_domain, AlgebraicGeometry.coprodMk_inl, AlgebraicGeometry.RingedSpace.zeroLocus_isClosed, AlgebraicGeometry.Scheme.Opens.toSpecΓ_naturality_assoc, AlgebraicGeometry.Scheme.Pullback.Triplet.hy, AlgebraicGeometry.LocallyRingedSpace.stalkMap_inv_hom_assoc, AlgebraicGeometry.IsAffineOpen.isoSpec_inv, AlgebraicGeometry.Scheme.Hom.id_app, AlgebraicGeometry.Scheme.Γ_obj, AlgebraicGeometry.isLocallyNoetherian_iff_of_affine_openCover, AlgebraicGeometry.Scheme.IdealSheafData.instIsEmptyCarrierCarrierCommRingCatSubschemeTop, AlgebraicGeometry.Scheme.Pullback.carrierEquiv_symm_fst, AlgHom.toUnder_comp, AlgebraicGeometry.Scheme.Hom.asFiberHom_fiberι_assoc, AlgebraicGeometry.instIsClosedUnderIsomorphismsSchemeIrreducibleSpaceCarrierCarrierCommRingCat, AlgebraicGeometry.Proj.mem_basicOpen, AlgebraicGeometry.IsLocallyArtinian.isArtinianRing_of_isAffine, AlgebraicGeometry.StructureSheaf.toPushforwardStalkAlgHom_apply, AlgebraicGeometry.Scheme.preimage_basicOpen_top, AlgebraicGeometry.Scheme.ΓSpecIso_inv, AlgebraicGeometry.Scheme.Hom.app_appIso_inv_assoc, pushoutCocone_inr, AlgebraicGeometry.IsAffineOpen.basicOpen_fromSpec_app, Localization.epi, AlgebraicGeometry.quasiCompact_over_affine_iff, AlgebraicGeometry.Scheme.Pullback.range_fst_comp, AlgebraicGeometry.Scheme.Hom.coequifibered_normalizationDiagramMap, AlgebraicGeometry.ΓSpec.adjunction_counit_app', AlgebraicGeometry.Scheme.isoSpec_hom_naturality_assoc, AlgebraicGeometry.Scheme.isoSpec_inv_image_zeroLocus, AlgebraicGeometry.Scheme.IdealSheafData.range_glueDataObjι_ι, AlgebraicGeometry.Scheme.germToFunctionField_injective, AlgebraicGeometry.Scheme.Hom.isIso_base, moduleCatRestrictScalarsPseudofunctor_mapId, instIsRightAdjointCommMonCatForget₂RingHomCarrierMonoidHomCarrier, AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction_counit_app', AlgebraicGeometry.Spec.topMap_id, AlgebraicGeometry.isIso_morphismRestrict_iff_isIso_app, AlgebraicGeometry.Proj.iSup_basicOpen_eq_top', AlgebraicGeometry.Scheme.hom_base_inv_base_assoc, AlgebraicGeometry.morphismRestrict_base, AlgebraicGeometry.Scheme.Opens.ι_image_basicOpen_topIso_inv, PresheafOfModules.Monoidal.tensorObj_map_tmul, AlgebraicGeometry.Scheme.zeroLocus_setMul, AlgebraicGeometry.SpecMap_ΓSpecIso_inv_toSpecΓ, AlgebraicGeometry.Scheme.Opens.nonempty_iff, AlgebraicGeometry.Scheme.toSpecΓ_preimage_zeroLocus, AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open, AlgebraicGeometry.Scheme.Hom.preimage_image_eq, AlgebraicGeometry.Scheme.Hom.quasiFiniteAt_comp_iff_of_isOpenImmersion, commAlgCatEquivUnder_functor_obj, TopCat.Presheaf.stalk_open_algebraMap, AlgebraicGeometry.isBasis_preimage_isAffineOpen, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.forgetToPresheafedSpace_reflectsPullback_of_right, AlgebraicGeometry.Scheme.Cover.isOpenEmbedding_fromGlued, AlgebraicGeometry.Scheme.Hom.app_surjective, AlgebraicGeometry.coe_opensRestrict_symm_apply, smoothSheafCommRing.forgetStalk_inv_comp_eval_assoc, AlgebraicGeometry.affineLocally_iff_affineOpens_le, AlgebraicGeometry.Scheme.fromSpecStalk_appTop, AlgebraicGeometry.RingedSpace.basicOpen_pow, AlgebraicGeometry.Scheme.OpenCover.isOpenCover_opensRange, AlgebraicGeometry.Scheme.Modules.toPresheaf_obj, AlgebraicGeometry.pullbackRestrictIsoRestrict_inv_fst, AlgebraicGeometry.Scheme.Hom.isCompact_preimage_singleton, AlgebraicGeometry.Scheme.Opens.toScheme_presheaf_obj, AlgebraicGeometry.Scheme.Modules.instFaithfulPresheafAbCarrierCommRingCatToPresheaf, AlgebraicGeometry.Scheme.IdealSheafData.vanishingIdeal_ideal, coproductCocone_inl, AlgebraicGeometry.Scheme.Pullback.Triplet.Spec_map_tensor_isPullback, Under.instPreservesFiniteLimitsUnderTensorProdOfFlatCarrier, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.invApp_app_apply, AlgebraicGeometry.Scheme.id_appTop, AlgebraicGeometry.Scheme.Cover.covers, AlgebraicGeometry.AffineScheme.Γ_preservesLimits, Colimits.cocone_naturality, AlgebraicGeometry.Scheme.Spec_map_presheaf_map_eqToHom, AlgebraicGeometry.Scheme.Opens.toSpecΓ_appTop_assoc, AlgebraicGeometry.Scheme.Hom.stalkMap_congr_point_assoc, AlgebraicGeometry.instIrreducibleSpaceCarrierCarrierCommRingCatSpecOfIsDomainCarrier, AlgebraicGeometry.Scheme.stalkMap_congr_point_assoc, smoothSheafCommRing.forgetStalk_inv_comp_eval, AlgebraicGeometry.Scheme.GlueData.isOpen_iff, AlgebraicGeometry.LocallyRingedSpace.stalkMap_congr_assoc, AlgebraicGeometry.IsAffineOpen.primeIdealOf_isMaximal_of_isClosed, AlgebraicGeometry.Scheme.fromSpecStalk_toSpecΓ, AlgebraicGeometry.Scheme.ker_toSpecΓ, AlgebraicGeometry.Scheme.IdealSheafData.ideal_bot, AlgebraicGeometry.Spec.toLocallyRingedSpace_obj, AlgebraicGeometry.Scheme.ker_of_isAffine, AlgebraicGeometry.IsAffineOpen.algebraMap_Spec_obj, AlgebraicGeometry.isBasis_basicOpen, HomTopology.precompHomeomorph_apply, AlgebraicGeometry.Scheme.Hom.mem_preimage, AlgebraicGeometry.Scheme.Spec_map_stalkMap_fromSpecStalk_assoc, AlgebraicGeometry.isInitial_iff_isEmpty, AlgebraicGeometry.LocallyRingedSpace.isLocalHomStalkMap, Under.instPreservesLimitsOfShapeUnderWalkingParallelPairTensorProdOfFlatCarrier, AlgebraicGeometry.IsImmersion.isLocallyClosed_range, AlgebraicGeometry.Scheme.Γevaluation_naturality_apply, AlgebraicGeometry.Scheme.basicOpen_le, AlgebraicGeometry.locallyOfFiniteType_iff, AlgebraicGeometry.Flat.flat_and_surjective_iff_faithfullyFlat_of_isAffine, AlgebraicGeometry.Scheme.Opens.toSpecΓ_SpecMap_presheaf_map_top_assoc, AlgebraicGeometry.liftCoborder_app, AlgebraicGeometry.LocallyRingedSpace.coe_toΓSpecSheafedSpace_hom_base_hom_apply_asIdeal, AlgebraicGeometry.Scheme.Modules.isSheaf, AlgebraicGeometry.Scheme.Opens.toSpecΓ_preimage_basicOpen, smoothSheafCommRing.forgetStalk_inv_comp_eval_apply, AlgebraicGeometry.isIntegral_appTop_of_universallyClosed, AlgebraicGeometry.Scheme.Cover.exists_lift_trans_eq, AlgebraicGeometry.Scheme.zeroLocus_def, AlgebraicGeometry.LocallyRingedSpace.toΓSpec_continuous, AlgebraicGeometry.Spec_map_localization_isIso, AlgebraicGeometry.StructureSheaf.isLocalizedModule_toPushforwardStalkAlgHom_aux, AlgebraicGeometry.Scheme.stalkClosedPointTo_fromSpecStalk, AlgebraicGeometry.IsOpenImmersion.map_ΓIso_inv_assoc, smoothSheafCommRing.isUnit_stalk_iff, AlgebraicGeometry.ΓSpec.adjunction_counit_app, AlgebraicGeometry.StructureSheaf.algebraMap_self_map, AlgebraicGeometry.Scheme.Hom.preimage_smoothLocus_eq, AlgebraicGeometry.tilde.isoTop_hom, AlgebraicGeometry.Scheme.coe_homeoOfIso_symm, AlgebraicGeometry.IsArtinianScheme.finite, AlgebraicGeometry.Spec.map_id, AlgebraicGeometry.ProjectiveSpectrum.Proj.isLocalization_atPrime, AlgebraicGeometry.instIrreducibleSpaceCarrierCarrierCommRingCatPullbackSchemeOfGeometricallyIrreducibleOfUniversallyOpen, AlgebraicGeometry.Scheme.stalkMap_hom_inv_assoc, ModuleCat.Tilde.toOpen_res, isPushout_tensorProduct, AlgebraicGeometry.IsLocallyArtinian.iff_isLocallyNoetherian_and_discreteTopology, AlgebraicGeometry.germ_comp_stalkToFiberRingHom, AlgebraicGeometry.Scheme.IdealSheafData.mem_supportSet_iff, AlgebraicGeometry.Scheme.id.base, AlgebraicGeometry.Scheme.IdealSheafData.range_subschemeι, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.scheme_toScheme, AlgebraicGeometry.Scheme.Opens.toSpecΓ_appTop, AlgebraicGeometry.Scheme.Γ_def, AlgebraicGeometry.Scheme.IdealSheafData.mem_support_iff, AlgebraicGeometry.Spec.full, AlgebraicGeometry.Scheme.IdealSheafData.le_support_iff_le_vanishingIdeal, AlgebraicGeometry.Scheme.Hom.isClosedMap, AlgebraicGeometry.Scheme.Hom.isLocallyClosed_range, mkUnder_hom, AlgebraicGeometry.ProjectiveSpectrum.Proj.toStalk_stalkMap_toSpec_assoc, AlgebraicGeometry.opensDiagram_map, AlgebraicGeometry.Scheme.residue_descResidueField, smoothSheafCommRing.ι_evalHom, AlgebraicGeometry.Γ_restrict_isLocalization, AlgebraicGeometry.Scheme.Hom.appIso_inv_app_presheafMap, AlgebraicGeometry.Scheme.Opens.toScheme_carrier, AlgebraicGeometry.IsClosedImmersion.iff_isPreimmersion, AlgEquiv.toUnder_trans, AlgebraicGeometry.Scheme.Hom.app_eq, AlgebraicGeometry.morphismRestrict_ι, AlgebraicGeometry.Scheme.Modules.restrictFunctorComp_hom_app_app, AlgebraicGeometry.Scheme.Hom.preimageIso_inv_ι, instIsLocalHomCarrierObjWalkingParallelPairFunctorConstPtEqualizerForkZeroParallelPairRingHomHomι, AlgebraicGeometry.Scheme.descResidueField_fromSpecResidueField, AlgebraicGeometry.ProjectiveSpectrum.Proj.toStalk_stalkMap_toSpec, moduleCatRestrictScalarsPseudofunctor_map, AlgebraicGeometry.Scheme.Hom.apply_mem_image_iff, AlgebraicGeometry.Scheme.Hom.asFiberHom_apply, AlgebraicGeometry.LocallyRingedSpace.stalkMap_id, AlgebraicGeometry.Spec.map_preimage_unop, RingEquiv.toCommRingCatIso_inv, AlgebraicGeometry.Scheme.Modules.restrictFunctorComp_inv_app_app, HomTopology.isEmbedding_pushout, AlgebraicGeometry.ΓSpecIso_obj_hom, coproductCocone_inr, AlgebraicGeometry.Scheme.SpecΓIdentity_app, AlgebraicGeometry.Proj.basicOpen_mul, AlgebraicGeometry.genericPoint_eq_of_isOpenImmersion, AlgebraicGeometry.Scheme.homeoOfIso_apply, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.inv_naturality_assoc, AlgebraicGeometry.Spec.germ_stalkMapIso_hom_assoc, AlgebraicGeometry.Scheme.IdealSheafData.le_ofIdeals_iff, AlgebraicGeometry.Scheme.Hom.preimage_basicOpen_top, forget_map, AlgebraicGeometry.StructureSheaf.IsLocalization.to_basicOpen, isPushout_of_isPushout, AlgebraicGeometry.Scheme.Modules.restrict_map, TopCat.Sheaf.objSupIsoProdEqLocus_hom_fst, AlgebraicGeometry.Scheme.PartialMap.restrict_id, AlgebraicGeometry.isIso_stalkMap_coprodSpec, AlgebraicGeometry.AffineSpace.spec_le_iff, AlgebraicGeometry.instPreservesColimitsOfShapeOppositeCommRingCatSchemeDiscreteSpecOfFinite, AlgebraicGeometry.Scheme.IsLocallyDirected.homOfLE_tAux, smoothSheafCommRing.evalHom_germ, AlgebraicGeometry.LocallyRingedSpace.residue_comp_residueFieldMap_eq_stalkMap_comp_residue, AlgebraicGeometry.HasRingHomProperty.iff_appLE, PresheafOfModules.Derivation.congr_d, AlgebraicGeometry.StructureSheaf.toOpen_comp_comap_assoc, AlgebraicGeometry.instQuasiSoberCarrierCarrierCommRingCat, AlgebraicGeometry.Scheme.Opens.exists_toScheme, AlgebraicGeometry.IsAffineOpen.self_le_basicOpen_union_iff, AlgebraicGeometry.exists_appTop_π_eq_of_isAffine_of_isLimit, AlgebraicGeometry.LocallyRingedSpace.GlueData.ι_isoSheafedSpace_inv, AlgebraicGeometry.Scheme.instIsIsoCommRingCatResidueFieldMapOfIsOpenImmersion, AlgebraicGeometry.Scheme.instIsOverMapResidueFieldMapOverInferInstanceOverClass, AlgebraicGeometry.Scheme.OpenCover.iSup_opensRange, AlgebraicGeometry.HasAffineProperty.restrict, Under.tensorProdEqualizer_ι, AlgebraicGeometry.instT0SpaceCarrierCarrierCommRingCat, AlgebraicGeometry.RingedSpace.mem_zeroLocus_iff, AlgebraicGeometry.Scheme.Γevaluation_naturality_assoc, AlgebraicGeometry.opensDiagramι_app, AlgebraicGeometry.Scheme.Pullback.carrierEquiv_symm_snd, AlgebraicGeometry.Scheme.IdealSheafData.coe_support_ofIdealTop, AlgebraicGeometry.Scheme.IdealSheafData.support_antitone, AlgebraicGeometry.Scheme.component_nontrivial, AlgebraicGeometry.isCompl_range_inl_inr, AlgebraicGeometry.Scheme.zeroLocus_iInf_of_nonempty, AlgebraicGeometry.Scheme.AffineZariskiSite.generate_presieveOfSections_mem_grothendieckTopology, AlgebraicGeometry.ΓSpec.toOpen_comp_locallyRingedSpaceAdjunction_homEquiv_app, AlgebraicGeometry.ΓSpec.adjunction_unit_app, AlgebraicGeometry.Scheme.basicOpen_pow, AlgebraicGeometry.Scheme.Opens.toSpecΓ_top, AlgebraicGeometry.LocallyRingedSpace.comp_base, AlgebraicGeometry.IsLocallyNoetherian.quasiSeparatedSpace, AlgebraicGeometry.Scheme.Hom.ker_apply, Under.instPreservesFiniteProductsUnderPushout, AlgebraicGeometry.Scheme.IdealSheafData.ideal_comap_of_isOpenImmersion, AlgebraicGeometry.Scheme.Pullback.range_map, AlgebraicGeometry.ProjIsoSpecTopComponent.fromSpec_toSpec, AlgebraicGeometry.SurjectiveOnStalks.isEmbedding_pullback, AlgebraicGeometry.AffineSpace.SpecIso_inv_appTop_coord, AlgebraicGeometry.germ_stalkClosedPointIso_hom_assoc, toAlgHom_id, AlgebraicGeometry.Scheme.Hom.isSpectralMap, HomTopology.instT2SpaceHomOfCarrier, AlgebraicGeometry.Scheme.IdealSheafData.support_eq_top_iff, AlgebraicGeometry.mem_range_iff_of_surjective, AlgebraicGeometry.Scheme.AffineZariskiSite.restrictIsoSpec_hom_app, smoothSheafCommRing.forgetStalk_hom_comp_evalHom_assoc, AlgebraicGeometry.instIsIsoSchemeMorphismRestrict, AlgebraicGeometry.Scheme.Pullback.Triplet.tensorCongr_trans, AlgebraicGeometry.Scheme.Modules.toPresheaf_map, AlgebraicGeometry.LocallyRingedSpace.restrictStalkIso_inv_eq_germ_apply, RingEquiv.toCommRingCatIso_hom, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.inv_invApp, hom_comp, AlgebraicGeometry.LocallyRingedSpace.stalkMap_inv_hom_apply, AlgebraicGeometry.Scheme.Hom.appIso_inv_app, AlgebraicGeometry.Scheme.Hom.liftCoborder_preimage, hasLimitsOfSize, HomTopology.precompHomeomorph_symm_apply, AlgebraicGeometry.Scheme.stalkMap_congr_hom_assoc, AlgebraicGeometry.Scheme.Hom.opensRange_pullbackFst, AlgebraicGeometry.Scheme.IdealSheafData.mem_supportSet_iff_mem_support, AlgebraicGeometry.Scheme.zeroLocus_radical, AlgebraicGeometry.Scheme.Modules.restrictFunctorCongr_inv_app_app, AlgebraicGeometry.pullbackRestrictIsoRestrict_hom_ι_assoc, mkUnder_right, AlgebraicGeometry.affineOpensRestrict_apply_coe_coe, AlgebraicGeometry.compactSpace_iff_quasiCompact, AlgebraicGeometry.Scheme.AffineZariskiSite.presieveOfSections_surjective, AlgebraicGeometry.RingedSpace.basicOpen_le, HomTopology.isClosedEmbedding_hom, AlgebraicGeometry.Scheme.Opens.ι_appIso, AlgebraicGeometry.IsAffineOpen.isoSpec_inv_toSpecΓ, AlgebraicGeometry.StructureSheaf.globalSectionsIso_hom, AlgebraicGeometry.disjoint_opensRange_sigmaι, AlgebraicGeometry.Scheme.comp_base_assoc, AlgebraicGeometry.instLocallyQuasiFiniteMorphismRestrict, AlgebraicGeometry.Scheme.stalkMap_id, AlgebraicGeometry.LocallyRingedSpace.comp_toShHom, AlgebraicGeometry.Scheme.Spec.residue_residueFieldIso_hom_assoc, AlgebraicGeometry.IsAffineOpen.toSpecΓ_isoSpec_inv, AlgebraicGeometry.Scheme.Hom.appIso_inv_naturality, hom_inv_apply, AlgebraicGeometry.LocallyRingedSpace.restrict_carrier, AlgebraicGeometry.Scheme.basicOpen_add_le, AlgebraicGeometry.Scheme.Hom.comp_app_assoc, AlgebraicGeometry.Scheme.comp_app, AlgebraicGeometry.Scheme.Hom.finite_app, AlgebraicGeometry.isIso_iff_isOpenImmersion, AlgebraicGeometry.Scheme.AffineZariskiSite.coequifibered_iff_forall_isLocalizationAway, AlgebraicGeometry.instGeometricallyReducedMorphismRestrict, AlgebraicGeometry.IsAffineOpen.fromSpec_preimage_zeroLocus, AlgebraicGeometry.Spec.sheafedSpaceMap_comp, AlgebraicGeometry.Scheme.GlueData.ι_eq_iff, AlgebraicGeometry.ΓSpec.left_triangle, AlgebraicGeometry.Scheme.Modules.toOpen_fromTildeΓ_app_assoc, AlgebraicGeometry.FormallyUnramified.instIsSeparableCarrierResidueFieldCoeContinuousMapCarrierCarrierCommRingCatHomTopCatBaseOfLocallyOfFiniteType, AlgebraicGeometry.Scheme.Hom.ideal_ker_le, AlgebraicGeometry.AffineSpace.comp_homOfVector, AlgebraicGeometry.smooth_iff, AlgebraicGeometry.Scheme.Hom.appIso_hom', AlgebraicGeometry.Scheme.zeroLocus_inf, AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction_homEquiv_apply', AlgebraicGeometry.AffineSpace.isOpenMap_over, AlgebraicGeometry.surjective_iff, AlgebraicGeometry.Scheme.IdealSheafData.ofIdeals_mono, AlgebraicGeometry.Scheme.instIsOpenImmersionToSpecΓOfIsQuasiAffine, epi_iff_tmul_eq_tmul, AlgebraicGeometry.Scheme.isoSpec_inv_naturality_assoc, AlgebraicGeometry.Spec.map_comp_assoc, AlgebraicGeometry.IsAffineOpen.fromSpec_app_self, AlgebraicGeometry.Spec.toPresheafedSpace_map_op, AlgebraicGeometry.IsOpenImmersion.opensEquiv_apply_coe, AlgebraicGeometry.isClosedImmersion_iff_isAffineHom, AlgebraicGeometry.instPreservesColimitsOfShapeOppositeCommRingCatSchemeDiscreteWalkingPairSpec, AlgebraicGeometry.isLocalIso_iff, AlgebraicGeometry.Spec.preimage_comp_assoc, AlgebraicGeometry.AffineSpace.hom_ext_iff, forget₂CommSemiRing_preservesLimits, AlgebraicGeometry.Scheme.SpecMap_stalkMap_fromSpecStalk_assoc, AlgebraicGeometry.ΓSpec.unop_locallyRingedSpaceAdjunction_counit_app', AlgebraicGeometry.Scheme.Hom.preimage_basicOpen, AlgebraicGeometry.Scheme.singleton_mem_precoverage_iff, AlgebraicGeometry.stalkMap_toStalk, AlgebraicGeometry.Scheme.IdealSheafData.subschemeι_app, AlgebraicGeometry.IsLocallyArtinian.discreteTopology, AlgebraicGeometry.Scheme.Hom.isOpenMap, AlgebraicGeometry.Scheme.PartialMap.equiv_iff_of_isSeparated, AlgEquiv.toUnder_inv_right_apply, AlgebraicGeometry.Scheme.Pullback.tensorCongr_SpecTensorTo_assoc, HomTopology.mvPolynomialHomeomorph_symm_apply_hom, AlgebraicGeometry.Scheme.isoSpec_Spec_hom, AlgebraicGeometry.Scheme.basicOpen_res_eq, AlgebraicGeometry.LocallyRingedSpace.instReflectsIsomorphismsSheafedSpaceCommRingCatForgetToSheafedSpace, AlgebraicGeometry.StructureSheaf.isLocalizedModule_toPushforwardStalkAlgHom, AlgebraicGeometry.Scheme.Pullback.Triplet.hx, AlgebraicGeometry.Scheme.Hom.isAffineOpen_iff_of_isOpenImmersion, AlgebraicGeometry.Scheme.Hom.stalkFunctor_toImage_injective, AlgebraicGeometry.isOpenImmersion_iff_stalk, AlgebraicGeometry.LocallyRingedSpace.Hom.ext_iff, AlgebraicGeometry.Scheme.image_basicOpen, AlgebraicGeometry.tilde.isIso_toOpen_top, AlgebraicGeometry.Scheme.instQuasiSeparatedSpaceCarrierCarrierCommRingCatOfIsSeparated, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.invApp_app, AlgebraicGeometry.LocallyRingedSpace.Hom.prop, TopCat.Presheaf.SubmonoidPresheaf.map, AlgebraicGeometry.SurjectiveOnStalks.surj_on_stalks, AlgebraicGeometry.Spec.fromSpecStalk_eq', AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.app_invApp_assoc, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.app_invApp, AlgebraicGeometry.HasRingHomProperty.appTop, AlgebraicGeometry.LocallyRingedSpace.basicOpen_zero, AlgebraicGeometry.isIntegral_iff_irreducibleSpace_and_isReduced, commMon_forget₂_obj_coe, AlgebraicGeometry.Spec.locallyRingedSpaceMap_comp, AlgebraicGeometry.IsAffineOpen.image_of_isOpenImmersion, toAlgHom_apply, AlgebraicGeometry.IsClosedImmersion.Spec_map_residue, AlgebraicGeometry.Scheme.isNilpotent_iff_basicOpen_eq_bot, AlgebraicGeometry.SurjectiveOnStalks.stalkMap_surjective, AlgebraicGeometry.IsIntegral.component_integral, AlgebraicGeometry.morphismRestrict_comp, pushout_inl_tensorProdObjIsoPushoutObj_inv_right_assoc, AlgebraicGeometry.IsProper.instMorphismRestrict, HomTopology.isEmbedding_hom, AlgebraicGeometry.Scheme.isPullback_toSpecΓ_toSpecΓ, AlgebraicGeometry.Scheme.Hom.genericPoint_mem_smoothLocus_of_perfectField, AlgebraicGeometry.Scheme.instSubsingletonCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensBot, AlgebraicGeometry.LocallyRingedSpace.restrictStalkIso_hom_eq_germ_apply, AlgebraicGeometry.IsArtinianScheme.toCompactSpace, moduleCatExtendScalarsPseudofunctor_map, TopCat.Presheaf.instMonoCommRingCatToTotalQuotientPresheafPresheaf, AlgebraicGeometry.Scheme.ΓSpecIso_naturality, AlgebraicGeometry.Scheme.Hom.stalkMap_hom_inv, AlgebraicGeometry.Scheme.IdealSheafData.support_comap, AlgebraicGeometry.Scheme.restrictFunctorΓ_hom_app, AlgebraicGeometry.Scheme.stalkMap_congr_hom, AlgebraicGeometry.Scheme.Modules.restrictFunctorCongr_hom_app_app, epi_iff_epi, hasLimits, AlgebraicGeometry.Scheme.IdealSheafData.vanishingIdeal_sup, AlgebraicGeometry.Scheme.Γ_map_op, AlgebraicGeometry.Scheme.Pullback.range_snd_comp, AlgebraicGeometry.instIsIsoSchemeAppUnitOppositeCommRingCatAdjunctionOfIsAffine, AlgebraicGeometry.Scheme.Modules.restrictFunctorId_inv_app_app, smoothSheafCommRing.ι_forgetStalk_inv_apply, AlgebraicGeometry.LocallyRingedSpace.GlueData.ι_jointly_surjective, AlgebraicGeometry.Scheme.Modules.pushforwardComp_hom_app_app, AlgebraicGeometry.instIsFractionRingCarrierStalkCommRingCatPresheafFunctionField, AlgebraicGeometry.Scheme.iso_inv_base_hom_base, AlgebraicGeometry.germ_stalkClosedPointIso_hom, AlgebraicGeometry.range_eq_univ, AlgebraicGeometry.Scheme.toSpecΓ_isoSpec_inv, AlgebraicGeometry.IsFinite.instHasAffinePropertyAndIsAffineFiniteCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTopHomAppTop, AlgebraicGeometry.SpecToEquivOfLocalRing_apply_fst, AlgebraicGeometry.Scheme.Hom.comp_apply, AlgebraicGeometry.IsAffineOpen.instIsAffineToSchemeBasicOpen, AlgebraicGeometry.Scheme.Modules.mapPresheaf_app, AlgebraicGeometry.Scheme.Hom.isOpenEmbedding, AlgebraicGeometry.opensDiagram_obj, coproductCocone_pt, AlgebraicGeometry.Scheme.Hom.preservesLocalization_normalizationDiagramMap, AlgebraicGeometry.Proj.basicOpen_mono, monoidAlgebra_map, AlgebraicGeometry.ΓSpec.isIso_locallyRingedSpaceAdjunction_counit, AlgebraicGeometry.tilde.instIsLocalizedModuleCarrierCarrierOfCarrierStalkAbPresheafPrimeComplAsIdealHomToStalk, AlgebraicGeometry.Scheme.residueFieldCongr_trans, AlgebraicGeometry.Spec.map_apply, forgetToRingCat_obj, Colimits.cocone_naturality_components, AlgebraicGeometry.pointOfClosedPoint_apply, AlgebraicGeometry.Spec.sheafedSpaceMap_hom_c_app, AlgebraicGeometry.Scheme.toIso_inv_ι_assoc, AlgebraicGeometry.morphismRestrict_app', AlgebraicGeometry.Spec.toSheafedSpace_obj, AlgebraicGeometry.Scheme.Hom.isEmbedding, AlgebraicGeometry.sigmaι_eq_iff, AlgebraicGeometry.opensCone_π_app, AlgebraicGeometry.Scheme.toSpecΓ_appTop, TopCat.Presheaf.epi_toLocalizationPresheaf, AlgebraicGeometry.Scheme.Hom.appIso_inv_naturality_assoc, AlgebraicGeometry.IsOpenImmersion.opensEquiv_symm_apply, AlgebraicGeometry.Scheme.IdealSheafData.support_eq_bot_iff, AlgebraicGeometry.GeometricallyIrreducible.irreducibleSpace_of_subsingleton, AlgebraicGeometry.Scheme.stalkMap_comp, smoothSheafCommRing.eval_surjective, AlgebraicGeometry.Scheme.residue_surjective, AlgebraicGeometry.instIsRightAdjointCommRingCatOppositeSchemeΓ, AlgebraicGeometry.isCompactOpen_iff_eq_finset_affine_union, AlgebraicGeometry.IsAffineOpen.iSup_basicOpen_eq_self_iff, AlgebraicGeometry.Scheme.Hom.comp_appIso, AlgebraicGeometry.instJacobsonSpaceCarrierCarrierCommRingCatSpecOfOfIsJacobsonRing, AlgebraicGeometry.Scheme.basicOpen_one, AlgebraicGeometry.IsOpenImmersion.range_pullback_to_base_of_left, AlgebraicGeometry.over_def, AlgebraicGeometry.Scheme.Modules.Hom.id_app, AlgebraicGeometry.Scheme.IdealSheafData.map_vanishingIdeal, AlgebraicGeometry.Scheme.Hom.stalkMap_inv_hom_assoc, AlgebraicGeometry.targetAffineLocally_affineAnd_iff, smoothSheafCommRing.forgetStalk_hom_comp_evalHom, AlgebraicGeometry.Scheme.Hom.appIso_inv_app_assoc, AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_stalk_isLocalHom, AlgebraicGeometry.IsAffineOpen.fromSpec_preimage_self, AlgebraicGeometry.Scheme.Γ_map, AlgebraicGeometry.Scheme.Hom.ker_eq_top_iff_isEmpty, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.forgetToTop_preservesPullback_of_left, AlgebraicGeometry.AffineScheme.forgetToScheme_map, AlgebraicGeometry.ProjectiveSpectrum.Proj.toSpec_base_apply_eq, AlgebraicGeometry.isClosed_singleton_iff_locallyOfFiniteType, AlgebraicGeometry.Scheme.Cover.ulift_I₀, FilteredColimits.forget_preservesFilteredColimits, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.instIsIsoCommRingCatInvApp, AlgebraicGeometry.LocallyRingedSpace.Γevaluation_naturality_apply, IsLocalization.epi, moduleCatExtendScalarsPseudofunctor_obj, AlgebraicGeometry.StructureSheaf.toStalk_stalkSpecializes_apply, commAlgCatEquivUnder_counitIso, AlgebraicGeometry.Scheme.Pullback.Triplet.tensorCongr_trans_hom_assoc, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.forgetToPresheafedSpace_preservesPullback_of_left, AlgebraicGeometry.Scheme.Modules.Hom.isIso_iff_isIso_app, AlgebraicGeometry.Scheme.Hom.residueFieldMap_congr, AlgebraicGeometry.Spec_zeroLocus_eq_zeroLocus, AlgebraicGeometry.Scheme.Opens.ι_appTop, AlgebraicGeometry.Scheme.Modules.Hom.comp_app, AlgebraicGeometry.Scheme.Hom.map_mem_image_iff, AlgebraicGeometry.Scheme.residueFieldCongr_fromSpecResidueField, AlgebraicGeometry.Scheme.IdealSheafData.support_iSup, AlgebraicGeometry.Scheme.coprodPresheafObjIso_hom_snd, RingHom.surjective_iff_epi_and_finite, AlgebraicGeometry.Scheme.map_basicOpen, AlgebraicGeometry.IsOpenImmersion.affineOpensEquiv_apply_coe_coe, isPushout_iff_isPushout, AlgebraicGeometry.coprodSpec_coprodMk, AlgebraicGeometry.Scheme.IdealSheafData.opensRange_subschemeCover_map, AlgebraicGeometry.Proj.zero_apply, PresheafOfModules.Derivation'.d_app, AlgebraicGeometry.isAffine_affineScheme, AlgebraicGeometry.isRetrocompact_basicOpen, AlgebraicGeometry.Scheme.Hom.preimageIso_hom_ι, AlgebraicGeometry.affinePreimage, AlgebraicGeometry.Scheme.Hom.injective, AlgebraicGeometry.isIso_iff_isOpenImmersion_and_epi_base, AlgebraicGeometry.quasiSeparatedSpace_of_isAffine, AlgebraicGeometry.Scheme.isoSpec_inv_naturality, AlgebraicGeometry.IsAffineOpen.toSpecΓ_fromSpec_assoc, AlgebraicGeometry.Spec_carrier, TopCat.Sheaf.objSupIsoProdEqLocus_inv_snd, AlgebraicGeometry.isArtinianScheme_iff, AlgebraicGeometry.Scheme.Hom.id_appIso, AlgebraicGeometry.Scheme.empty_carrier_carrier, smoothSheafCommRing.ι_forgetStalk_inv_assoc, AlgebraicGeometry.IsArtinianScheme.iff_isNoetherian_and_discreteTopology, AlgebraicGeometry.Scheme.Hom.preimageIso_inv_ι_assoc, AlgebraicGeometry.Scheme.Hom.quasiFiniteLocus_eq_top_iff, AlgebraicGeometry.instQuasiSeparatedToSpecΓOfQuasiSeparatedSpaceCarrierCarrierCommRingCat, AlgebraicGeometry.IsAffineOpen.basicOpenSectionsToAffine_isIso, AlgebraicGeometry.isAffineHom_iff, AlgebraicGeometry.Scheme.Hom.asFiberHom_fiberι, pushout_inl_tensorProdObjIsoPushoutObj_inv_right, AlgebraicGeometry.exists_smooth_of_formallySmooth_stalk, AlgebraicGeometry.Scheme.comp_appTop_assoc, AlgebraicGeometry.QuasiCompact.compactSpace_of_compactSpace, AlgebraicGeometry.Scheme.local_affine, AlgebraicGeometry.Proj.basicOpen_one, AlgebraicGeometry.Scheme.IdealSheafData.ideal_top, AlgebraicGeometry.quasiSeparated_iff_quasiSeparatedSpace, commAlgCatEquivUnder_inverse_obj_carrier, AlgebraicGeometry.Spec.locallyRingedSpaceMap_toHom, TopCat.Presheaf.instIsLocalizationCarrierObjOppositeOpensCarrierCommRingCatObjLocalizationPresheaf, AlgebraicGeometry.LocallyOfFiniteType.stalkMap, ModuleCat.Derivation.d_map, AlgebraicGeometry.instIsAffineObjOppositeCommRingCatSchemeSpec, Under.instPreservesLimitUnderDiscreteFunctorTensorProdOfFinite, AlgebraicGeometry.locallyQuasiFinite_iff_isDiscrete_preimage_singleton, AlgebraicGeometry.Scheme.IsLocallyDirected.ι_eq_ι_iff, AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalHom, ofHom_apply, AlgebraicGeometry.Scheme.isoSpec_Spec, AlgebraicGeometry.ProjectiveSpectrum.Proj.toSpec_base_isIso, AlgebraicGeometry.instLocallyOfFiniteTypeMorphismRestrict, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.locallyRingedSpace_toLocallyRingedSpace, AlgebraicGeometry.IsIntegralHom.instMorphismRestrict, AlgebraicGeometry.isCompact_and_isOpen_iff_finite_and_eq_biUnion_affineOpens, AlgebraicGeometry.Scheme.Pullback.Triplet.specTensorTo_base_fst, AlgebraicGeometry.compactSpace_iff_exists, AlgebraicGeometry.instIrreducibleSpaceCarrierCarrierCommRingCatFiberOfGeometricallyIrreducible, AlgebraicGeometry.Scheme.Hom.preimage_sup, AlgebraicGeometry.LocallyRingedSpace.ofRestrict_stalkMap_isIso, AlgebraicGeometry.IsClosedImmersion.hasAffineProperty, AlgebraicGeometry.Scheme.preimage_opensRange_toSpecΓ, AlgebraicGeometry.instIsOpenImmersionMorphismRestrict, AlgebraicGeometry.Scheme.Spec.algebraMap_residueFieldIso_inv, AlgebraicGeometry.Scheme.iso_hom_base_inv_base, AlgebraicGeometry.Scheme.stalkMap_congr_point, AlgHom.toUnder_right, AlgebraicGeometry.ProjectiveSpectrum.Proj.mk_mem_toSpec_base_apply, AlgebraicGeometry.Scheme.eq_zeroLocus_of_isClosed_of_isAffine, AlgebraicGeometry.Scheme.Opens.ι_preimage_self, AlgebraicGeometry.Scheme.stalkMap_hom_inv, AlgebraicGeometry.Scheme.residueFieldCongr_refl, AlgebraicGeometry.Scheme.toSpecΓ_image_zeroLocus, AlgebraicGeometry.instQuasiSeparatedMorphismRestrict, AlgebraicGeometry.Scheme.Hom.image_iSup, AlgebraicGeometry.LocallyRingedSpace.evaluation_naturality, AlgebraicGeometry.isImmersion_iff, AlgebraicGeometry.quasiSeparatedSpace_iff_forall_affineOpens, PresheafOfModules.DifferentialsConstruction.relativeDifferentials'_map_d, AlgebraicGeometry.instNonemptyCarrierCarrierCommRingCatSpecOfNontrivialCarrier, AlgebraicGeometry.toSpecΓ_SpecMap_ΓSpecIso_inv_assoc, AlgEquiv.toUnder_hom_right_apply, AlgebraicGeometry.Scheme.Hom.comp_preimage, AlgebraicGeometry.Scheme.Modules.instIsIsoAbApp, AlgebraicGeometry.Scheme.Hom.naturality_assoc, AlgebraicGeometry.Scheme.IdealSheafData.opensRange_glueDataObjMap, AlgebraicGeometry.IsAffineOpen.fromSpec_app_self_apply, AlgebraicGeometry.Scheme.stalkMap_hom_inv_apply, AlgebraicGeometry.Scheme.Hom.iInf_ker_openCover_map_comp_apply, HomTopology.continuous_apply, AlgebraicGeometry.Scheme.Hom.isOpen_quasiFiniteAt, preservesFilteredColimits_coyoneda, AlgebraicGeometry.Scheme.inv_appTop, AlgebraicGeometry.isLocalization_away_of_isAffine, AlgebraicGeometry.isPullback_Spec_map_pushout, AlgebraicGeometry.IsAffineOpen.ι_basicOpen_preimage, AlgebraicGeometry.Scheme.Hom.quasiFiniteAt_iff, AlgebraicGeometry.instIsRightAdjointCommRingCatOppositeLocallyRingedSpaceΓ, AlgebraicGeometry.LocallyRingedSpace.isLocalHomStalkMap', AlgebraicGeometry.LocallyRingedSpace.stalkMap_hom_inv, forget_map_apply, AlgebraicGeometry.morphismRestrict_appTop, AlgebraicGeometry.Scheme.Hom.exists_mem_and_isIso_morphismRestrict_toNormalization, AlgebraicGeometry.Scheme.IdealSheafData.ofIdealTop_ideal, AlgebraicGeometry.GeometricallyIrreducible.eq_geometrically, AlgebraicGeometry.Proj.basicOpen_eq_iSup_proj, commAlgCatEquivUnder_unitIso, isFinitelyPresentable_under, AlgebraicGeometry.Scheme.Hom.preimage_opensRange, AlgebraicGeometry.ΓSpecIso_hom_stalkClosedPointIso_inv, AlgebraicGeometry.AffineSpace.toSpecMvPolyIntEquiv_apply, AlgebraicGeometry.Proj.isBasis_basicOpen, AlgebraicGeometry.Scheme.descResidueField_stalkClosedPointTo_fromSpecResidueField, forget₂Ring_preservesLimits, AlgebraicGeometry.IsAffineOpen.fromSpec_preimage_basicOpen, AlgebraicGeometry.LocallyRingedSpace.Γ_map, AlgebraicGeometry.isCompactOpen_iff_eq_basicOpen_union, AlgebraicGeometry.Spec.algebraMap_stalkIso_inv, AlgebraicGeometry.geometricallyIrreducible_iff, AlgebraicGeometry.isDominant_iff, AlgebraicGeometry.Flat.generalizingMap, AlgebraicGeometry.Scheme.Hom.fiberι_asFiber, AlgebraicGeometry.AffineSpace.isIntegralHom_over_iff_isEmpty, AlgebraicGeometry.HasAffineProperty.affineAnd_iff, AlgebraicGeometry.Scheme.default_asIdeal, AlgebraicGeometry.IsOpenImmersion.app_eq_appIso_inv_app_of_comp_eq, AlgebraicGeometry.Scheme.Hom.id_base, AlgebraicGeometry.Scheme.IdealSheafData.ker_subschemeι_app, AlgebraicGeometry.IsOpenImmersion.ΓIso_inv, AlgebraicGeometry.Scheme.Modules.Hom.sub_app, AlgebraicGeometry.IsOpenImmersion.range_pullback_snd_of_left, AlgebraicGeometry.LocallyRingedSpace.restrict_presheaf_map, AlgebraicGeometry.AffineScheme.Spec_essSurj, AlgebraicGeometry.IsOpenImmersion.instIsIsoCommRingCatStalkMap, AlgebraicGeometry.isPreimmersion_iff, AlgebraicGeometry.Scheme.AffineCover.covers, AlgebraicGeometry.ΓSpec.adjunction_homEquiv_apply, AlgebraicGeometry.Proj.add_apply, AlgebraicGeometry.Scheme.Hom.ι_toNormalization_assoc, equalizer_ι_isLocalHom', AlgebraicGeometry.Scheme.Hom.Spec_map_residueFieldMap_fromSpecResidueField_assoc, AlgebraicGeometry.Scheme.isLocalHom_stalkClosedPointTo', AlgebraicGeometry.IsFinite.finite_preimage_singleton, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.forgetToPresheafedSpacePreservesOpenImmersion, AlgebraicGeometry.IsLocalAtTarget.restrict, AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.mk_mem_carrier, AlgebraicGeometry.Scheme.Hom.fromNormalization_app_assoc, AlgebraicGeometry.Scheme.isoSpec_inv_toSpecΓ_assoc, FilteredColimits.forget₂Ring_preservesFilteredColimits, AlgebraicGeometry.ProjectiveSpectrum.Proj.isIso_toSpec, AlgebraicGeometry.Scheme.IdealSheafData.ideal_iInf, AlgebraicGeometry.stalkMap_toStalk_apply, AlgebraicGeometry.LocallyRingedSpace.forgetToTop_map, AlgebraicGeometry.IsAffineOpen.primeIdealOf_eq_map_closedPoint, AlgebraicGeometry.Scheme.Hom.isoImage_inv_ι, AlgebraicGeometry.Scheme.GlueData.ι_isoCarrier_inv, AlgebraicGeometry.Scheme.IdealSheafData.range_glueDataObjι, AlgebraicGeometry.Scheme.Hom.isoImage_inv_ι_assoc, AlgebraicGeometry.Scheme.Hom.instIsIsoCommRingCatAppObjOpensOpensFunctor, AlgebraicGeometry.LocallyRingedSpace.homOfSheafedSpaceHomOfIsIso_toHom, RingHom.surjective_of_epi_of_finite, AlgebraicGeometry.Scheme.ker_morphismRestrict_ideal, AlgebraicGeometry.Scheme.Modules.pushforwardId_hom_app_app, AlgebraicGeometry.LocallyRingedSpace.instIsLocalRingCarrierStalkCommRingCatPresheaf, AlgebraicGeometry.pullbackRestrictIsoRestrict_hom_ι, localization_unit_isIso, AlgebraicGeometry.Scheme.coprodPresheafObjIso_hom_fst, AlgebraicGeometry.Scheme.toOpen_eq, AlgebraicGeometry.IsOpenImmersion.range_pullback_to_base_of_right, AlgebraicGeometry.Spec.toSheafedSpace_map, AlgebraicGeometry.Scheme.Modules.restrict_obj, AlgebraicGeometry.IsOpenImmersion.app_ΓIso_hom_assoc, PresheafOfModules.DifferentialsConstruction.relativeDifferentials'_obj, AlgebraicGeometry.Scheme.Modules.restrictAdjunction_unit_app_app, AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction_homEquiv_apply, AlgebraicGeometry.isCompl_opensRange_inl_inr, TopCat.Presheaf.restrictOpenCommRingCat_apply, AlgebraicGeometry.Scheme.Hom.isIntegral_app, AlgebraicGeometry.Scheme.Hom.app_eq_appLE, AlgebraicGeometry.Scheme.Pullback.Triplet.fst_SpecTensorTo_apply, AlgebraicGeometry.ΓSpecIso_inv_ΓSpec_adjunction_homEquiv, AlgebraicGeometry.RingedSpace.zeroLocus_empty_eq_univ, AlgebraicGeometry.StructureSheaf.algebraMap_germ_assoc, TopCat.Presheaf.SubmonoidPresheaf.toLocalizationPresheaf_app, AlgebraicGeometry.Scheme.Modules.Hom.add_app, PresheafOfModules.Derivation.d_app, AlgebraicGeometry.Scheme.IdealSheafData.isClosed_supportSet, AlgebraicGeometry.toSpecΓ_SpecMap_ΓSpecIso_inv, AlgebraicGeometry.IsOpenImmersion.map_ΓIso_inv, AlgebraicGeometry.isBasis_affine_open, AlgebraicGeometry.instIsIsoModulesSpecOfCarrierFromTildeΓUnitOpensCarrierCarrierCommRingCatRingCatSheaf, AlgebraicGeometry.IsClosedImmersion.isClosedEmbedding, AlgebraicGeometry.LocallyRingedSpace.GlueData.ι_isoSheafedSpace_inv_assoc, AlgebraicGeometry.RingedSpace.basicOpen_mul, AlgebraicGeometry.Scheme.Hom.inv_appTop, coproductCoconeIsColimit_desc, AlgebraicGeometry.IsLocallyArtinian.discreteTopology_of_isAffine, AlgebraicGeometry.isIntegralHom_iff, AlgebraicGeometry.RingedSpace.basicOpen_res_eq, AlgebraicGeometry.Spec_zeroLocus, AlgebraicGeometry.Scheme.basicOpen_mul, AlgebraicGeometry.Scheme.ΓSpecIso_inv_naturality, AlgebraicGeometry.Scheme.IdealSheafData.coe_support_eq_eq_iInter_zeroLocus, PresheafOfModules.Monoidal.tensorHom_app, PresheafOfModules.DifferentialsConstruction.relativeDifferentials'_map, AlgebraicGeometry.Spec.topMap_comp, AlgebraicGeometry.Spec.map_comp, AlgebraicGeometry.Scheme.AffineZariskiSite.instIsDenseSubsiteOpensCarrierCarrierCommRingCatGrothendieckTopologyGrothendieckTopologyToOpensFunctor, tensorProdIsoPushout_app, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.isoRestrict_inv_ofRestrict_assoc, inv_hom_apply, AlgebraicGeometry.Scheme.zeroLocus_biInf_of_nonempty, AlgebraicGeometry.Flat.stalkMap, AlgebraicGeometry.Scheme.Pullback.range_snd, AlgebraicGeometry.Scheme.Hom.stalkMap_comp, AlgebraicGeometry.Scheme.SpecΓIdentity_inv_app, AlgebraicGeometry.LocallyRingedSpace.toΓSpec_base, AlgebraicGeometry.instJacobsonSpaceCarrierCarrierCommRingCatFiberOfLocallyOfFiniteType, AlgebraicGeometry.StructureSheaf.algebraMap_germ, AlgebraicGeometry.Scheme.Cover.exists_eq, AlgebraicGeometry.IsOpenImmersion.app_ΓIso_hom, AlgebraicGeometry.preservesLimit_rightOp_Γ, AlgebraicGeometry.Scheme.ofArrows_mem_precoverage_iff, AlgebraicGeometry.Scheme.GlueData.instIsOpenImmersionCommRingCatFPresheafedSpaceToPresheafedSpaceGlueDataToSheafedSpaceGlueDataToLocallyRingedSpaceGlueData, AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction_unit, PresheafOfModules.Derivation'.app_apply, comp_apply, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toSchemeHom_toPshHom, AlgebraicGeometry.StructureSheaf.instIsLocalizedModuleCarrierStalkAbPresheafOpensCarrierTopModuleStructurePresheafPrimeComplAsIdealToStalkₗ, AlgebraicGeometry.Spec.homEquiv_apply, AlgebraicGeometry.Scheme.IsLocallyDirected.ι_jointly_surjective, AlgebraicGeometry.basicOpen_eq_of_affine', AlgebraicGeometry.IsPreimmersion.base_embedding, RingHom.toMorphismProperty_respectsIso_iff, AlgebraicGeometry.Scheme.Hom.coe_image, AlgebraicGeometry.AffineSpace.isoOfIsAffine_hom_appTop, AlgebraicGeometry.Scheme.Hom.coe_preimage, AlgebraicGeometry.LocallyRingedSpace.stalkSpecializes_stalkMap_apply, AlgebraicGeometry.Spec.coe_toTop_map_hom_apply_asIdeal, AlgebraicGeometry.Scheme.Opens.toScheme_presheaf_map, AlgebraicGeometry.instIsDominantToSpecΓOfCompactSpaceCarrierCarrierCommRingCat, smoothSheafCommRing.ι_forgetStalk_hom_apply, TopCat.Sheaf.objSupIsoProdEqLocus_hom_snd, AlgebraicGeometry.stalkToFiberRingHom_germ, AlgebraicGeometry.LocallyRingedSpace.restrictStalkIso_inv_eq_ofRestrict, AlgebraicGeometry.Scheme.isoSpec_Spec_inv, AlgebraicGeometry.Scheme.IdealSheafData.supportSet_subset_zeroLocus, AlgebraicGeometry.instFaithfulOppositeCommRingCatLocallyRingedSpaceToLocallyRingedSpace, AlgebraicGeometry.Spec.map_eqToHom, AlgebraicGeometry.Scheme.Hom.isClosedEmbedding, AlgebraicGeometry.Proj.awayι_preimage_basicOpen, AlgebraicGeometry.Scheme.Hom.quasiFiniteAt_iff_isOpen_singleton_asFiber, AlgebraicGeometry.isClosed_singleton_iff_isClosedImmersion, TopCat.Sheaf.objSupIsoProdEqLocus_inv_eq_iff, AlgebraicGeometry.IsFinite.finite_app, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toLocallyRingedSpaceHom_val, AlgebraicGeometry.instSubsingletonCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensOfIsEmpty, AlgebraicGeometry.exists_eq_pow_mul_of_isAffineOpen, AlgebraicGeometry.Scheme.Opens.forall_toScheme, smoothSheafCommRing.ι_forgetStalk_hom_assoc, AlgebraicGeometry.Scheme.topIso_hom, AlgebraicGeometry.Proj.awayMap_awayToSection, AlgebraicGeometry.Scheme.Hom.finite_preimage_singleton, AlgebraicGeometry.instPreservesColimitsOfShapeOppositeCommRingCatSchemeDiscretePEmptySpec, AlgebraicGeometry.finite_appTop_of_universallyClosed, AlgebraicGeometry.Scheme.AffineZariskiSite.toOpens_mono, AlgebraicGeometry.IsLocallyNoetherian.component_noetherian, AlgebraicGeometry.Scheme.inv_hom_apply, AlgebraicGeometry.instIsPreimmersionFromSpecStalk, AlgebraicGeometry.IsIntegralHom.integral_app, commAlgCatEquivUnder_inverse_map, AlgebraicGeometry.isIso_SpecMap_stakMap_localization, AlgebraicGeometry.Scheme.residue_residueFieldMap_assoc, coyonedaUnique_inv_app_hom_apply, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.invApp_app_assoc, AlgebraicGeometry.Scheme.Spec_fromSpecStalk', AlgebraicGeometry.Scheme.IdealSheafData.support_map, AlgebraicGeometry.FormallyUnramified.stalkMap, AlgebraicGeometry.specTargetImageFactorization_app_injective, AlgebraicGeometry.Scheme.IsGermInjectiveAt.cond, AlgebraicGeometry.Scheme.IdealSheafData.zeroLocus_inter_subset_supportSet, AlgebraicGeometry.IsAffineOpen.preimage_of_isIso, AlgebraicGeometry.instIrreducibleSpaceCarrierCarrierCommRingCatPullbackSchemeOfGeometricallyIrreducibleOfUniversallyOpen_1, AlgebraicGeometry.instIsClosedImmersionMorphismRestrict, AlgebraicGeometry.Scheme.Hom.preimage_inf, AlgebraicGeometry.ΓSpec.right_triangle, AlgebraicGeometry.isPullback_morphismRestrict, AlgebraicGeometry.Scheme.IdealSheafData.map_ideal', AlgebraicGeometry.Scheme.residue_residueFieldCongr, AlgebraicGeometry.StructureSheaf.comap_id_eq_map, AlgebraicGeometry.Scheme.Hom.appIso_hom, AlgebraicGeometry.Scheme.OpenCover.finiteSubcover_X, AlgebraicGeometry.quasiSeparatedSpace_of_quasiSeparated, AlgebraicGeometry.Scheme.Pullback.range_fst, Under.instPreservesFiniteProductsUnderTensorProd, AlgebraicGeometry.IsOpenImmersion.iff_stalk_iso, AlgebraicGeometry.Scheme.Cover.fromGlued_isOpenEmbedding, AlgebraicGeometry.IsOpenImmersion.app_ΓIso_hom_apply, AlgebraicGeometry.Scheme.residue_residueFieldMap, AlgebraicGeometry.Scheme.IdealSheafData.ideal_sup, AlgebraicGeometry.LocallyRingedSpace.isLocalRing, AlgebraicGeometry.AffineSpace.isoOfIsAffine_hom, AlgebraicGeometry.StructureSheaf.stalkAlgebra_map, AlgebraicGeometry.tilde.toOpen_res, AlgebraicGeometry.tilde.toOpen_res_assoc, AlgebraicGeometry.Scheme.Hom.ι_fromNormalization_assoc, AlgebraicGeometry.LocallyRingedSpace.iso_inv_base_hom_base, equalizer_ι_isLocalHom, AlgebraicGeometry.Scheme.emptyTo_c_app, AlgebraicGeometry.Scheme.stalkMap_inv_hom_apply, AlgebraicGeometry.instCompactSpaceCarrierCarrierCommRingCatFiberOfQuasiCompact, AlgebraicGeometry.Scheme.IsQuasiAffine.toCompactSpace, AlgebraicGeometry.IsLocalIso.exists_isOpenImmersion, AlgebraicGeometry.Scheme.Opens.mem_basicOpen_toScheme, AlgebraicGeometry.Scheme.Hom.comp_base_assoc, AlgebraicGeometry.affineAnd_apply, AlgebraicGeometry.Scheme.SpecMap_residue_apply, AlgebraicGeometry.tilde.isUnit_algebraMap_end_basicOpen, AlgebraicGeometry.HasRingHomProperty.iff_exists_appLE_locally, preservesColimit_coyoneda_of_finitePresentation, AlgebraicGeometry.Scheme.Hom.image_le_opensRange, AlgebraicGeometry.noetherianSpace_of_isAffineOpen, AlgebraicGeometry.Scheme.Pullback.range_diagonal_subset_diagonalCoverDiagonalRange, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.forgetToPresheafedSpace_reflectsPullback_of_left, 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, AlgebraicGeometry.Scheme.instIsPreimmersionMapResidue, AlgebraicGeometry.Scheme.Hom.mem_smoothLocus, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.forget_reflectsPullback_of_left, AlgebraicGeometry.ΓSpec.adjunction_homEquiv_symm_apply, AlgebraicGeometry.instSmoothMorphismRestrict, AlgebraicGeometry.Spec_Γ_naturality_assoc, AlgebraicGeometry.Scheme.IdealSheafData.ideal_mono, AlgebraicGeometry.RingedSpace.isUnit_res_basicOpen, AlgebraicGeometry.Scheme.empty_presheaf, AlgebraicGeometry.IsAffineOpen.isoSpec_hom_apply, AlgebraicGeometry.Scheme.Opens.ι_app_self, AlgebraicGeometry.ΓSpec.toSpecΓ_of, AlgebraicGeometry.Scheme.zeroLocus_span, AlgebraicGeometry.StructureSheaf.toOpen_germ, AlgebraicGeometry.Scheme.IdealSheafData.ideal_iSup, AlgebraicGeometry.Scheme.Hom.naturality, AlgebraicGeometry.quasiSeparatedSpace_iff_quasiSeparated, AlgebraicGeometry.Scheme.Hom.image_injective, HomTopology.isHomeomorph_precomp, AlgebraicGeometry.Proj.stalkIso'_germ, AlgebraicGeometry.exists_isFinite_morphismRestrict_of_finite_preimage_singleton, AlgebraicGeometry.IsAffine.affine, AlgebraicGeometry.IsAffineOpen.instAwayCarrierObjOppositeOpensCarrierCarrierCommRingCatSpecPresheafOpOpensBasicOpen, AlgebraicGeometry.AffineTargetMorphismProperty.IsLocal.to_basicOpen, AlgebraicGeometry.Scheme.Pullback.Triplet.Spec_ofPointTensor_SpecTensorTo, Colimits.hasColimits_commRingCat, AlgebraicGeometry.Scheme.IdealSheafData.vanishingIdeal_top, AlgebraicGeometry.Proj.basicOpen_zero, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.isoRestrict_hom_ofRestrict, AlgebraicGeometry.formallyUnramified_iff, AlgebraicGeometry.Spec.locallyRingedSpaceObj_presheaf_map', AlgebraicGeometry.Scheme.Pullback.Triplet.ofPoint_s, AlgebraicGeometry.Scheme.mem_basicOpen_top, moduleCatExtendScalarsPseudofunctor_mapComp, AlgebraicGeometry.QuasiCompactCover.isCompactOpenCovered_of_isAffineOpen, coyonedaUnique_hom_app_hom_apply, AlgebraicGeometry.Scheme.Opens.range_ι, AlgebraicGeometry.IsAffineOpen.range_fromSpec, AlgebraicGeometry.Spec.map_inv, AlgebraicGeometry.IsOpenImmersion.image_preimage_eq_preimage_image_of_isPullback, AlgebraicGeometry.isClosedImmersion_iff, AlgebraicGeometry.Scheme.Opens.ι_image_basicOpen', AlgebraicGeometry.LocallyRingedSpace.Γevaluation_naturality_assoc, AlgebraicGeometry.Spec.sheafedSpaceMap_hom_base, AlgebraicGeometry.Spec_closedPoint, AlgebraicGeometry.Scheme.Hom.range_subset_ker_support, AlgebraicGeometry.LocallyRingedSpace.preimage_basicOpen, AlgebraicGeometry.ProjectiveSpectrum.Proj.stalkMap_toSpec, AlgebraicGeometry.LocallyRingedSpace.is_sheafedSpace_iso, AlgebraicGeometry.IsAffineOpen.isoSpec_inv_toSpecΓ_assoc, AlgebraicGeometry.isPullback_SpecMap_pushout, AlgebraicGeometry.AffineSpace.homOverEquiv_apply, AlgebraicGeometry.Scheme.Hom.finite_preimage, AlgebraicGeometry.Scheme.Opens.ι_apply, AlgebraicGeometry.Scheme.Modules.instFullPresheafOfModulesToPresheafOfModules, AlgebraicGeometry.Scheme.Hom.dense_smoothLocus_of_perfectField, AlgebraicGeometry.quasiCompact_iff_compactSpace, AlgebraicGeometry.pullbackRestrictIsoRestrict_inv_fst_assoc, AlgebraicGeometry.Scheme.basicOpen_res, AlgebraicGeometry.Scheme.iso_inv_base_hom_base_apply, AlgebraicGeometry.StructureSheaf.toPushforwardStalk_comp, AlgebraicGeometry.Scheme.Hom.continuous, AlgebraicGeometry.Scheme.Hom.asFiberHom_fiberToSpecResidueField, TopCat.Sheaf.objSupIsoProdEqLocus_inv_fst, AlgebraicGeometry.Smooth.exists_isStandardSmooth, AlgebraicGeometry.Scheme.Pullback.Triplet.exists_preimage, AlgebraicGeometry.Scheme.comp_appTop, AlgebraicGeometry.Scheme.Hom.stalkMap_hom_inv_apply, smoothSheafCommRing.ι_evalHom_assoc, AlgebraicGeometry.Surjective.surj, AlgebraicGeometry.Scheme.IdealSheafData.radical_ideal, HomTopology.mvPolynomialHomeomorph_apply_fst, AlgebraicGeometry.isCompact_iff_finite_and_eq_biUnion_affineOpens, AlgebraicGeometry.IsClosedImmersion.isAffine_surjective_of_isAffine, AlgebraicGeometry.Scheme.AffineZariskiSite.mem_grothendieckTopology, instIsRightAdjointOppositeObjFunctorTypeYoneda, Under.equalizerFork_ι, AlgebraicGeometry.Scheme.stalkMap_inv_hom, AlgebraicGeometry.StructureSheaf.toPushforwardStalk_comp_assoc, AlgebraicGeometry.Spec.toPresheafedSpace_obj_op, AlgebraicGeometry.Scheme.comp_base_apply, AlgebraicGeometry.isGermInjectiveAt_iff_of_isOpenImmersion, AlgebraicGeometry.SpecMap_residueFieldIsoBase_inv, AlgebraicGeometry.IsAffineOpen.isLocalization_stalk, AlgebraicGeometry.AffineSpace.toSpecMvPolyIntEquiv_comp, AlgebraicGeometry.SpecMap_residueFieldIsoBase_inv_assoc, AlgebraicGeometry.isField_of_universallyClosed, AlgebraicGeometry.IsAffineOpen.isoSpec_hom_base_apply, AlgebraicGeometry.Scheme.IdealSheafData.ideal_inf, AlgebraicGeometry.LocallyRingedSpace.Γevaluation_eq_zero_iff_notMem_basicOpen, AlgebraicGeometry.ProjectiveSpectrum.Proj.toSpec_base_apply_eq_comap, AlgebraicGeometry.Scheme.Spec.algebraMap_residueFieldIso_inv_assoc, smoothSheafCommRing.ι_forgetStalk_hom, AlgebraicGeometry.IsDominant.denseRange, AlgebraicGeometry.Spec.sheafedSpaceMap_id, AlgebraicGeometry.ProjectiveSpectrum.Proj.awayToSection_germ, AlgebraicGeometry.instGeometricallyIntegralMorphismRestrict, AlgebraicGeometry.Flat.iff_flat_stalkMap, AlgebraicGeometry.Scheme.IdealSheafData.subschemeι_app_surjective, AlgebraicGeometry.Scheme.preimage_zeroLocus, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.isoRestrict_hom_ofRestrict_assoc, AlgebraicGeometry.isIso_iff_stalk_iso, AlgebraicGeometry.Scheme.instIsOverMapStalkMapOverInferInstanceOverClass, AlgebraicGeometry.Scheme.AffineZariskiSite.cocone_ι_app, AlgebraicGeometry.Scheme.Hom.smoothLocus_eq_top, AlgebraicGeometry.Spec.map_base, AlgebraicGeometry.Scheme.instIsOverMapHomCommRingCatResidueFieldCongr, AlgebraicGeometry.Scheme.IdealSheafData.coe_support_inter, AlgebraicGeometry.compactSpace_of_universallyClosed, AlgebraicGeometry.IsAffineOpen.fromSpec_top, AlgebraicGeometry.Scheme.Hom.support_ker, commAlgCatEquivUnder_functor_map, AlgebraicGeometry.SpecToEquivOfLocalRing_symm_apply, piFan_pt, AlgebraicGeometry.Scheme.Hom.homeomorph_apply, AlgebraicGeometry.Scheme.Hom.stalkMap_congr, AlgebraicGeometry.Scheme.Hom.isIso_toPshHom, AlgebraicGeometry.Scheme.IdealSheafData.support_top, AlgebraicGeometry.Scheme.map_basicOpen_map, AlgebraicGeometry.affineOpensRestrict_symm_apply_coe, AlgebraicGeometry.Scheme.Hom.comp_appTop, AlgebraicGeometry.instNoetherianSpaceCarrierCarrierCommRingCatSpecOfIsNoetherianRingCarrier, instIsLeftAdjointCommMonCatUnderMonoidAlgebra, AlgebraicGeometry.IsOpenImmersion.opensRange_pullback_snd_of_left, AlgebraicGeometry.IsIntegralHom.hasAffineProperty, AlgebraicGeometry.instJacobsonSpaceCarrierCarrierCommRingCatSpecOfIsJacobsonRingCarrier, AlgebraicGeometry.Scheme.germ_stalkClosedPointTo_Spec, AlgebraicGeometry.LocallyRingedSpace.stalkSpecializes_stalkMap, AlgebraicGeometry.Scheme.basicOpen_zero, AlgebraicGeometry.Scheme.Hom.surjective, AlgebraicGeometry.Scheme.residue_residueFieldCongr_assoc, AlgebraicGeometry.Scheme.IdealSheafData.ideal_ofIdeals_le, monoidAlgebra_obj, AlgebraicGeometry.Scheme.IsLocallyDirected.exists_of_pullback_V_V, AlgebraicGeometry.Scheme.SpecMap_presheaf_map_eqToHom, AlgebraicGeometry.IsAffineOpen.fromSpec_image_basicOpen, AlgebraicGeometry.instHasAffinePropertyIsomorphismsSchemeAndIsAffineIsIsoCommRingCatAppTop, AlgebraicGeometry.pullbackRestrictIsoRestrict_hom_morphismRestrict, AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage, AlgebraicGeometry.IsAffineOpen.basicOpen_union_eq_self_iff, AlgebraicGeometry.LocallyRingedSpace.comp_toHom, AlgebraicGeometry.LocallyRingedSpace.restrictStalkIso_hom_eq_germ_assoc, AlgebraicGeometry.isFinite_iff, AlgebraicGeometry.etale_iff, CategoryTheory.Iso.commRingCatIsoToRingEquiv_toRingHom, AlgebraicGeometry.isAffineOpen_bot, AlgebraicGeometry.Scheme.IsLocallyDirected.V_self, AlgebraicGeometry.LocallyRingedSpace.residueFieldMap_id, AlgebraicGeometry.Scheme.Opens.ι_image_le, AlgebraicGeometry.Spec.toPresheafedSpace_map, AlgebraicGeometry.StructureSheaf.comap_basicOpen, AlgebraicGeometry.Scheme.Hom.stalkMap_congr_assoc, AlgebraicGeometry.LocallyRingedSpace.forgetToTop_obj, AlgebraicGeometry.IsOpenImmersion.opensRange_pullback_fst_of_right, AlgebraicGeometry.Scheme.mem_basicOpen'', AlgebraicGeometry.stalkClosedPointIso_inv, AlgebraicGeometry.Scheme.AffineZariskiSite.cocone_pt, AlgebraicGeometry.specTargetImageRingHom_surjective, AlgebraicGeometry.isEmpty_pullback_sigmaι_of_ne, AlgebraicGeometry.Scheme.stalkMap_inv_hom_assoc, AlgebraicGeometry.Scheme.Spec_obj, AlgebraicGeometry.Scheme.Cover.RelativeGluingData.preimage_toBase_eq_range_ι, AlgebraicGeometry.tilde.toOpen_map_app_assoc, AlgebraicGeometry.LocallyRingedSpace.forgetToSheafedSpace_map, AlgebraicGeometry.locallyQuasiFinite_iff_finite_preimage_singleton, AlgebraicGeometry.IsClosedImmersion.SpecMap_residue, AlgebraicGeometry.LocallyRingedSpace.Γevaluation_ne_zero_iff_mem_basicOpen, AlgebraicGeometry.basicOpen_eq_of_affine, AlgebraicGeometry.LocallyRingedSpace.Γ_map_op, AlgebraicGeometry.AffineScheme.forgetToScheme_obj, AlgebraicGeometry.Scheme.ΓSpecIso_naturality_assoc, AlgebraicGeometry.IsAffineOpen.fromSpec_toSpecΓ_assoc, AlgebraicGeometry.Scheme.Hom.comp_image, AlgebraicGeometry.IsOpenImmersion.range_pullbackFst, hom_id, AlgebraicGeometry.IsAffineOpen.fromSpec_image_zeroLocus, AlgebraicGeometry.Spec.toLocallyRingedSpace_map, isLocalHom_of_iso, AlgebraicGeometry.Scheme.Cover.instEpiTopCatBaseCommRingCatFromGlued, AlgebraicGeometry.LocallyRingedSpace.isUnit_res_toΓSpecMapBasicOpen, AlgebraicGeometry.Scheme.affineBasisCover_map_range, AlgebraicGeometry.Scheme.residue_descResidueField_assoc, AlgebraicGeometry.ΓSpec_adjunction_homEquiv_eq, AlgebraicGeometry.Flat.isQuotientMap_of_surjective, AlgebraicGeometry.Scheme.Opens.ι_app, AlgebraicGeometry.Scheme.Hom.QuasiFiniteAt.isClopen_singleton_asFiber, AlgebraicGeometry.Scheme.Modules.pushforwardCongr_inv_app_app, AlgebraicGeometry.Scheme.IdealSheafData.supportSet_inter, AlgebraicGeometry.instIsPreimmersionAsFiberHom, AlgebraicGeometry.locallyOfFinitePresentation_iff, AlgebraicGeometry.Scheme.Hom.isoImage_hom_ι, AlgebraicGeometry.IsAffineOpen.isOpenImmersion_fromSpec, AlgebraicGeometry.LocallyRingedSpace.stalkMap_congr_hom_assoc, AlgebraicGeometry.quasiCompact_iff_spectral, AlgebraicGeometry.Scheme.iso_hom_base_inv_base_apply, commMon_forget₂_map, AlgebraicGeometry.instUniversallyClosedMorphismRestrict, AlgebraicGeometry.Scheme.toIso_inv_ι, AlgebraicGeometry.quasiCompact_iff_isSpectralMap, AlgebraicGeometry.Scheme.Hom.instIsIsoCommRingCatApp, AlgebraicGeometry.Scheme.Spec_map, AlgebraicGeometry.Scheme.residueFieldCongr_trans_hom, AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec_preimage_basicOpen, AlgebraicGeometry.Scheme.Hom.inv_image, AlgebraicGeometry.Scheme.residueFieldCongr_fromSpecResidueField_assoc, isLocalHom_comp, AlgebraicGeometry.IsPreimmersion.instMorphismRestrict, AlgebraicGeometry.Scheme.Hom.opensRange_comp, AlgebraicGeometry.Scheme.Hom.ι_fromNormalization, AlgebraicGeometry.LocallyRingedSpace.toΓSpec_preimage_zeroLocus_eq, AlgebraicGeometry.LocallyRingedSpace.restrictStalkIso_inv_eq_germ, AlgebraicGeometry.Scheme.Hom.comp_base, AlgebraicGeometry.exists_finite_imageι_comp_morphismRestrict_of_finite_image_preimage, AlgebraicGeometry.genericPoint_eq_bot_of_affine, AlgebraicGeometry.LocallyRingedSpace.SpecΓIdentity_hom_app, AlgebraicGeometry.IsAffineOpen.isCompactOpenCovered, AlgebraicGeometry.instIsQuasicoherentOpensCarrierCarrierCommRingCatSpecTilde, AlgebraicGeometry.IsOpenImmersion.map_ΓIso_hom, AlgebraicGeometry.LocallyRingedSpace.evaluation_ne_zero_iff_mem_basicOpen, AlgebraicGeometry.Scheme.IdealSheafData.coe_support_vanishingIdeal, AlgebraicGeometry.Scheme.Γevaluation_naturality, AlgebraicGeometry.Scheme.residueFieldMap_id, AlgebraicGeometry.IsSeparated.instMorphismRestrict, AlgebraicGeometry.Scheme.PartialMap.dense_domain, AlgebraicGeometry.basicOpen_eq_bot_iff, AlgebraicGeometry.Scheme.Modules.instReflectsIsomorphismsPresheafAbCarrierCommRingCatToPresheaf, AlgebraicGeometry.Scheme.IdealSheafData.ker_glueDataObjι_appTop, AlgebraicGeometry.Spec.germ_stalkMapIso_hom, AlgebraicGeometry.Scheme.Hom.appIso_inv_app_apply, AlgebraicGeometry.HasRingHomProperty.stalkMap_of_respectsIso, localization_unit_isIso', AlgebraicGeometry.Scheme.restrictFunctor_obj_left, isFinitelyPresentable, AlgebraicGeometry.LocallyOfFiniteType.jacobsonSpace, AlgebraicGeometry.Spec.fromSpecStalk_eq, AlgebraicGeometry.quasiCompact_iff_forall_affine, AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv_over, AlgebraicGeometry.Spec.locallyRingedSpaceObj_presheaf, AlgebraicGeometry.Scheme.Hom.stalkMap_id, AlgebraicGeometry.instLocallyOfFinitePresentationMorphismRestrict, moduleCatExtendScalarsPseudofunctor_mapId, AlgebraicGeometry.Scheme.IdealSheafData.ideal_biInf, AlgebraicGeometry.Scheme.isLocalHom_stalkClosedPointTo, AlgebraicGeometry.LocallyRingedSpace.Γevaluation_naturality, AlgebraicGeometry.Proj.iSup_basicOpen_eq_top, AlgebraicGeometry.IsAffineOpen.fromSpec_preimage_basicOpen', AlgebraicGeometry.Scheme.basicOpen_restrict, AlgebraicGeometry.structurePresheafInCommRingCat_obj_carrier, AlgebraicGeometry.Scheme.IdealSheafData.support_bot, ofHom_comp, AlgebraicGeometry.Scheme.Hom.opensRange_of_isIso, AlgebraicGeometry.IsAffineOpen.app_basicOpen_eq_away_map, AlgebraicGeometry.LocallyRingedSpace.toΓSpecMapBasicOpen_eq, forget_preservesLimits, AlgebraicGeometry.Scheme.preimage_basicOpen, AlgebraicGeometry.Scheme.Opens.instIsIsoCommRingCatAppLEιTopToScheme, AlgebraicGeometry.IsAffineHom.isAffine_preimage, AlgebraicGeometry.IsAffineOpen.ΓSpecIso_hom_fromSpec_app, AlgebraicGeometry.SpecToEquivOfLocalRing_eq_iff, AlgebraicGeometry.Scheme.restrictFunctor_map_left, AlgebraicGeometry.Scheme.Hom.SpecMap_residueFieldMap_fromSpecResidueField, AlgebraicGeometry.Scheme.affineOpenCover_X, AlgebraicGeometry.Scheme.comp_app_assoc, AlgebraicGeometry.IsAffineOpen.isQuasiSeparated, AlgebraicGeometry.Scheme.Pullback.Triplet.ofPoint_y, AlgebraicGeometry.instQuasiCompactToSpecΓOfCompactSpaceCarrierCarrierCommRingCat, AlgebraicGeometry.LocallyRingedSpace.evaluation_naturality_assoc, AlgebraicGeometry.morphismRestrict_app, AlgebraicGeometry.AffineSpace.comp_homOfVector_assoc, forgetToRingCat_map_hom, AlgebraicGeometry.Scheme.PartialMap.le_domain_toRationalMap, AlgebraicGeometry.LocallyRingedSpace.toStalk_stalkMap_toΓSpec, AlgebraicGeometry.Scheme.Modules.instIsRightAdjointPresheafOfModulesToPresheafOfModules, AlgebraicGeometry.Scheme.Opens.ι_image_top, AlgebraicGeometry.Scheme.Hom.toImage_app, AlgebraicGeometry.Scheme.range_fromSpecResidueField, AlgebraicGeometry.Scheme.Modules.restrictFunctorId_hom_app_app, AlgebraicGeometry.Scheme.SpecMap_stalkMap_fromSpecStalk, AlgebraicGeometry.LocallyRingedSpace.id_toHom, AlgebraicGeometry.Scheme.zeroLocus_isClosed, AlgebraicGeometry.Scheme.AffineZariskiSite.instIsLocallyFullOpensToOpensFunctorGrothendieckTopologyCarrierCarrierCommRingCat, AlgebraicGeometry.IsAffineOpen.isoSpec_hom, AlgebraicGeometry.Scheme.Modules.Hom.app_smul, AlgebraicGeometry.LocallyRingedSpace.toΓSpecSheafedSpace_app_spec, AlgebraicGeometry.Scheme.Γ_obj_op, AlgebraicGeometry.Scheme.Hom.stalkMap_inv_hom_apply, AlgebraicGeometry.Scheme.Hom.mem_quasiFiniteLocus, AlgebraicGeometry.quasiCompactCover_iff, AlgebraicGeometry.Scheme.image_zeroLocus, AlgebraicGeometry.Scheme.basicOpen_eq_bot_iff_forall_evaluation_eq_zero, AlgebraicGeometry.HasRingHomProperty.iff_of_source_openCover, AlgebraicGeometry.Scheme.Hom.isoImage_hom_ι_assoc, AlgebraicGeometry.LocallyRingedSpace.toΓSpecCBasicOpens_app, AlgebraicGeometry.LocallyRingedSpace.evaluation_eq_zero_iff_notMem_basicOpen, AlgebraicGeometry.isIso_toSpecΓ, AlgebraicGeometry.Proj.res_apply, AlgebraicGeometry.Scheme.isBasis_affineOpens, AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.preimage_basicOpen, AlgebraicGeometry.Scheme.Hom.fiberToSpecResidueField_apply, AlgebraicGeometry.Scheme.inv_base_hom_base_assoc, AlgebraicGeometry.Scheme.Opens.topIso_hom, AlgebraicGeometry.Scheme.Hom.stalkMap_hom_inv_assoc, AlgebraicGeometry.Scheme.IdealSheafData.support_sup, AlgebraicGeometry.StructureSheaf.toOpen_res, AlgebraicGeometry.LocallyRingedSpace.toΓSpecSheafedSpace_app_spec_assoc, AlgebraicGeometry.Scheme.IsLocallyDirected.homOfLE_tAux_assoc, AlgebraicGeometry.Scheme.Pullback.Triplet.ofPoint_SpecTensorTo, AlgebraicGeometry.Scheme.OpenCover.restrict_X, PresheafOfModules.germ_smul, AlgebraicGeometry.Scheme.zeroLocus_univ, AlgebraicGeometry.LocallyRingedSpace.toΓSpecCApp_spec, AlgebraicGeometry.Scheme.JointlySurjective.exists_eq, AlgebraicGeometry.Scheme.Hom.image_le_image_iff, AlgebraicGeometry.Scheme.Hom.comp_appTop_assoc, AlgebraicGeometry.AffineSpace.map_appTop_coord, AlgebraicGeometry.Scheme.toSpecΓ_preimage_basicOpen, AlgebraicGeometry.LocallyRingedSpace.isLocalHom_stalkMap_congr, prodFan_pt, AlgebraicGeometry.Scheme.residueFieldCongr_trans_hom_assoc, AlgebraicGeometry.Scheme.Hom.app_appIso_inv, AlgebraicGeometry.isIso_fromTildeΓ_iff, AlgebraicGeometry.Scheme.residueFieldCongr_inv, AlgebraicGeometry.Scheme.coe_homeoOfIso, AlgebraicGeometry.LocallyRingedSpace.stalkSpecializes_stalkMap_assoc, AlgebraicGeometry.Scheme.Hom.normalizationObjIso_hom_val, AlgebraicGeometry.Scheme.openCoverBasicOpenTop_f, AlgebraicGeometry.pointEquivClosedPoint_symm_apply_coe, AlgebraicGeometry.LocallyRingedSpace.toΓSpecSheafedSpace_hom_c_app, AlgebraicGeometry.Scheme.Hom.appLE_eq_app, moduleCatRestrictScalarsPseudofunctor_obj, AlgebraicGeometry.Scheme.RationalMap.mem_domain, AlgebraicGeometry.Scheme.IdealSheafData.support_sSup, AlgebraicGeometry.instHasAffinePropertyQuasiCompactCompactSpaceCarrierCarrierCommRingCat, AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec.image_basicOpen_eq_basicOpen, smoothSheafCommRing.forgetStalk_hom_comp_evalHom_apply, AlgebraicGeometry.IsOpenImmersion.app_eq_invApp_app_of_comp_eq_aux, AlgebraicGeometry.isField_of_isIntegral_of_subsingleton, AlgebraicGeometry.spec_punit_isEmpty, AlgebraicGeometry.LocallyRingedSpace.instPreservesColimitsOfShapeSheafedSpaceCommRingCatDiscreteForgetToSheafedSpace, AlgebraicGeometry.isClosedMap_iff_specializingMap, AlgebraicGeometry.instIsNoetherianRingCarrierStalkCommRingCatPresheafOfIsLocallyNoetherian, AlgebraicGeometry.isReduced_stalk_of_isReduced, AlgebraicGeometry.Scheme.image_preimage_eq_of_isPullback, AlgebraicGeometry.ProjectiveSpectrum.Proj.toOpen_toSpec_val_c_app, AlgebraicGeometry.Scheme.Hom.instIsLocalHomCarrierStalkCommRingCatPresheafCoeContinuousMapCarrierCarrierHomTopCatBaseRingHomHomStalkMap, TopCat.Presheaf.submonoidPresheafOfStalk_obj, AlgebraicGeometry.LocallyRingedSpace.stalkMap_comp, AlgebraicGeometry.Scheme.GlueData.ι_jointly_surjective, Under.instPreservesLimitUnderWalkingParallelPairParallelPairTensorProdOfFlatCarrier, AlgebraicGeometry.StructureSheaf.algebraMap_pushforward_stalk, AlgebraicGeometry.Scheme.IdealSheafData.vanishingIdeal_iSup, AlgebraicGeometry.Scheme.IdealSheafData.strictMono_ideal, AlgebraicGeometry.LocallyRingedSpace.stalkMap_hom_inv_apply, AlgebraicGeometry.Scheme.codisjoint_zeroLocus, AlgebraicGeometry.LocallyRingedSpace.forgetToSheafedSpace_obj, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.forget_reflectsPullback_of_right, AlgebraicGeometry.Scheme.PartialMap.restrict_id_hom, AlgebraicGeometry.Scheme.stalkClosedPointTo_comp, AlgebraicGeometry.LocallyRingedSpace.evaluation_naturality_apply, AlgebraicGeometry.IsAffineOpen.fromSpec_toSpecΓ, AlgebraicGeometry.map_injective_of_isIntegral, AlgebraicGeometry.AffineSpace.homOfVector_appTop_coord, AlgebraicGeometry.Scheme.Spec_map_residue_apply, AlgebraicGeometry.sourceAffineLocally_morphismRestrict, AlgebraicGeometry.Scheme.IsQuasiAffine.isBasis_basicOpen, AlgebraicGeometry.initial_isEmpty, AlgebraicGeometry.Proj.stalkIso'_symm_mk, AlgebraicGeometry.Scheme.residueFieldMap_comp, AlgebraicGeometry.Etale.instMorphismRestrict, CommAlgCat.forget₂_commRingCat_map, AlgebraicGeometry.Scheme.isoSpec_image_zeroLocus, AlgebraicGeometry.Scheme.Hom.fiberι_fiberHomeo_symm, AlgebraicGeometry.Scheme.comp_coeBase, AlgebraicGeometry.Scheme.IdealSheafData.ideal_map_of_isAffineHom, AlgebraicGeometry.Scheme.Spec_stalkClosedPointTo_fromSpecStalk, AlgebraicGeometry.isCompact_and_isOpen_iff_finite_and_eq_biUnion_basicOpen, AlgebraicGeometry.LocallyRingedSpace.iso_hom_base_inv_base, AlgebraicGeometry.LocallyRingedSpace.isLocalHomValStalkMap, AlgebraicGeometry.localRingHom_comp_stalkIso, AlgebraicGeometry.IsAffineOpen.toSpecΓ_isoSpec_inv_assoc, AlgebraicGeometry.Scheme.Modules.pushforward_map_app, AlgebraicGeometry.LocallyRingedSpace.SpecΓIdentity_inv_app, AlgebraicGeometry.Scheme.mem_toGrothendieck_smallPretopology, AlgebraicGeometry.LocallyRingedSpace.stalkMap_congr_point, AlgebraicGeometry.Scheme.Hom.stalkMap_inv_hom, AlgebraicGeometry.coe_opensRestrict_apply_coe, AlgebraicGeometry.Scheme.AffineZariskiSite.presieveOfSections_eq_ofArrows, AlgebraicGeometry.Spec.toTop_obj_carrier, AlgebraicGeometry.Scheme.Hom.inv_app, AlgebraicGeometry.Scheme.localRingHom_comp_stalkIso, AlgebraicGeometry.Scheme.restrictFunctor_obj_hom, AlgebraicGeometry.finite_irreducibleComponents_of_isNoetherian, AlgebraicGeometry.Scheme.Hom.app_injective, AlgebraicGeometry.Scheme.Modules.inv_app, AlgebraicGeometry.Scheme.Hom.Spec_map_residueFieldMap_fromSpecResidueField, AlgebraicGeometry.Scheme.Opens.toSpecΓ_preimage_zeroLocus, AlgebraicGeometry.Scheme.Cover.fromGlued_open_map, AlgebraicGeometry.AffineScheme.Spec_full, preservesFilteredColimits_coyoneda, AlgebraicGeometry.coprodMk_inr, AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec_bijective, AlgebraicGeometry.LocallyRingedSpace.comp_c_app, AlgebraicGeometry.Scheme.instEpiCommRingCatResidue, AlgebraicGeometry.Scheme.Hom.isProperMap, AlgebraicGeometry.Scheme.isoOfEq_hom, AlgebraicGeometry.LocallyRingedSpace.toΓSpec_preimage_basicOpen_eq, AlgebraicGeometry.StructureSheaf.algebraMap_germ_apply, AlgebraicGeometry.Scheme.Pullback.Triplet.tensorCongr_symm, AlgebraicGeometry.Scheme.le_iff_specializes, AlgebraicGeometry.Scheme.Hom.asFiberHom_fiberToSpecResidueField_assoc, AlgebraicGeometry.quasiSeparatedSpace_iff_quasiCompact_prod_lift, AlgebraicGeometry.Scheme.Spec.residue_residueFieldIso_hom, AlgebraicGeometry.Scheme.IdealSheafData.vanishingIdeal_sSup, AlgebraicGeometry.Scheme.AffineZariskiSite.mem_grothendieckTopology_iff_sectionsOfPresieve, AlgebraicGeometry.IsAffineOpen.isoSpec_inv_ι, AlgebraicGeometry.IsAffineOpen.isoSpec_hom_appTop, AlgebraicGeometry.Scheme.IdealSheafData.map_ideal_basicOpen, AlgebraicGeometry.Scheme.AffineZariskiSite.toOpensFunctor_obj, coproductCocone_ι, AlgebraicGeometry.Scheme.localRingHom_comp_stalkIso_apply, AlgebraicGeometry.instIsIsoModulesSpecOfCarrierFromTildeΓFreeOpensCarrierCarrierCommRingCat, AlgebraicGeometry.RingedSpace.zeroLocus_singleton, AlgebraicGeometry.Scheme.Pullback.Triplet.tensorCongr_refl, AlgebraicGeometry.LocallyRingedSpace.stalkMap_congr_hom, preservesColimit_coyoneda_of_finitePresentation, AlgebraicGeometry.essImage_Spec, AlgebraicGeometry.Scheme.Hom.isOpen_smoothLocus, AlgebraicGeometry.Scheme.mem_zeroLocus_iff, AlgebraicGeometry.Scheme.Hom.fromNormalization_app, AlgebraicGeometry.Spec_stalkClosedPointIso, AlgebraicGeometry.Scheme.Pullback.Triplet.tensorCongr_inv, AlgebraicGeometry.Scheme.Opens.ι_base_apply, AlgebraicGeometry.Scheme.Hom.eqToHom_app, AlgebraicGeometry.instIsAffineHomιBasicOpen, AlgebraicGeometry.LocallyRingedSpace.preservesCoequalizer, AlgebraicGeometry.Scheme.id_app, AlgebraicGeometry.flat_iff, AlgebraicGeometry.Scheme.zeroLocus_map_of_eq, AlgebraicGeometry.IsAffineOpen.toSpecΓ_fromSpec, AlgebraicGeometry.Scheme.residueFieldCongr_symm, AlgebraicGeometry.Scheme.PartialMap.fromSpecStalkOfMem_toPartialMap, AlgebraicGeometry.IsAffineOpen.fromSpec_primeIdealOf, AlgebraicGeometry.LocallyRingedSpace.GlueData.instPreservesLimitSheafedSpaceCommRingCatWalkingCospanCospanFForgetToSheafedSpace, AlgebraicGeometry.LocallyRingedSpace.iso_hom_base_inv_base_apply, AlgebraicGeometry.Scheme.homOfLE_rfl, AlgebraicGeometry.Scheme.fromSpecResidueField_apply, AlgebraicGeometry.Scheme.AffineZariskiSite.instIsCoverDenseOpensToOpensFunctorGrothendieckTopologyCarrierCarrierCommRingCat, AlgebraicGeometry.IsAffineOpen.isoSpec_inv_ι_assoc, AlgebraicGeometry.HasRingHomProperty.iff_of_isAffine, instIsRightAdjointForgetRingHomCarrier, AlgebraicGeometry.Scheme.Hom.coe_opensRange, AlgebraicGeometry.quasiSeparatedSpace_iff_affine, AlgebraicGeometry.Scheme.IdealSheafData.le_def, AlgebraicGeometry.Scheme.stalkMap_congr, AlgebraicGeometry.Spec.homEquiv_symm_apply, AlgebraicGeometry.Scheme.AffineZariskiSite.restrictIsoSpec_inv_app, AlgebraicGeometry.ΓSpec.toSpecΓ_unop, AlgebraicGeometry.Spec.locallyRingedSpaceObj_sheaf, AlgebraicGeometry.isClosedImmersion_diagonal_restrict_diagonalCoverDiagonalRange, Under.equalizer_comp, AlgebraicGeometry.Scheme.Hom.stalkMap_congr_hom, RingHom.IsStableUnderBaseChange.pushout_inl, AlgebraicGeometry.AffineScheme.ΓIsEquiv, AlgebraicGeometry.Scheme.inv_base_hom_base, AlgebraicGeometry.Scheme.IdealSheafData.vanishingIdeal_bot, AlgebraicGeometry.tilde.instIsLocalizedModuleCarrierCarrierObjOppositeOpensCarrierCarrierCommRingCatSpecModuleCatPresheafModulesSheafModulesSpecToSheafOpBasicOpenPowersHomToOpen, AlgebraicGeometry.instIsOpenImmersionCommRingCatOfIsOpenImmersion, AlgebraicGeometry.quasiSeparated_over_affine_iff, Under.equalizerFork'_ι, AlgebraicGeometry.Scheme.IdealSheafData.subschemeι_apply, AlgebraicGeometry.Scheme.IdealSheafData.supportSet_eq_iInter_zeroLocus, AlgebraicGeometry.Scheme.Modules.map_smul, AlgebraicGeometry.RingedSpace.mem_basicOpen', AlgebraicGeometry.IsOpenImmersion.affineOpensEquiv_symm_apply_coe, AlgebraicGeometry.Scheme.affineBasicOpen_le, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.isoRestrict_inv_ofRestrict, isPushout_of_isLocalization, AlgebraicGeometry.Scheme.Spec_stalkClosedPointTo_fromSpecStalk_assoc, AlgebraicGeometry.SpecMap_ΓSpecIso_inv_toSpecΓ_assoc, AlgebraicGeometry.IsOpenImmersion.map_ΓIso_inv_apply, AlgebraicGeometry.LocallyRingedSpace.toΓSpecCApp_iff, AlgebraicGeometry.IsAffineOpen.preimage, AlgebraicGeometry.Spec.locallyRingedSpaceObj_sheaf', AlgebraicGeometry.LocallyRingedSpace.comp_c, AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction_counit, AlgebraicGeometry.LocallyRingedSpace.Γ_obj, AlgebraicGeometry.Spec.sheafedSpaceObj_presheaf, AlgebraicGeometry.Spec.map_app, AlgebraicGeometry.Scheme.Spec_fromSpecStalk, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.forget_preservesPullbackOfLeft, AlgebraicGeometry.homogeneousLocalizationToStalk_stalkToFiberRingHom, AlgebraicGeometry.Scheme.Cover.fromGlued_injective, AlgebraicGeometry.IsAffineOpen.ideal_ext_iff, AlgebraicGeometry.Spec.map_base_apply, AlgebraicGeometry.morphismRestrict_id, AlgebraicGeometry.Scheme.Hom.stalkMap_surjective, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.forget_preservesPullback_of_right, AlgebraicGeometry.tilde.toOpen_map_app, AlgebraicGeometry.Spec.preimage_comp, AlgebraicGeometry.IsAffineOpen.primeIdealOf_genericPoint, pushout_inr_tensorProdObjIsoPushoutObj_inv_right, AlgebraicGeometry.Scheme.Hom.id_image, AlgebraicGeometry.IsAffineOpen.mem_ideal_iff, AlgebraicGeometry.Scheme.toSpecΓ_base, AlgebraicGeometry.Smooth.iff_forall_exists_isStandardSmooth
instCoeSortType 📖	CompOp	—
instCommRingObjForgetRingHomCarrier 📖	CompOp	—
instConcreteCategoryRingHomCarrier 📖	CompOp	198 math math: PresheafOfModules.Monoidal.tensorObj_obj, smoothSheafCommRing.ι_forgetStalk_inv, AlgebraicGeometry.Scheme.coprodPresheafObjIso_hom_fst_assoc, forget₂Ring_preservesLimitsOfSize, KaehlerDifferential.map_d, AlgebraicGeometry.StructureSheaf.instIsScalarTowerCarrierStalkCommRingCatStructurePresheafInCommRingCatCarrierAbPresheafOpensCarrierTopModuleStructurePresheaf, HomTopology.mvPolynomialHomeomorph_apply_snd, instFullRingCatForget₂RingHomCarrierCarrier, forget_preservesLimitsOfSize, forgetReflectIsos, AlgebraicGeometry.RingedSpace.basicOpen_res, AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv, closure_range_union_range_eq_top_of_isPushout, AlgebraicGeometry.surjectiveOnStalks_iff, AlgebraicGeometry.LocallyRingedSpace.notMem_prime_iff_unit_in_stalk, AlgebraicGeometry.LocallyRingedSpace.basicOpen_eq_bot_iff_forall_evaluation_eq_zero, AlgebraicGeometry.Scheme.mem_basicOpen', id_apply, smoothSheafCommRing.ι_evalHom_apply, AlgebraicGeometry.SpecMap_preimage_basicOpen, mkUnder_ext_iff, smoothSheafCommRing.eval_germ, forget₂CommSemiRing_preservesLimitsOfSize, AlgebraicGeometry.localRingHom_comp_stalkIso_apply, CommAlgCat.forget₂_commRingCat_obj, forget_obj, PresheafOfModules.Derivation.d_map, free_map_coe, AlgebraicGeometry.Scheme.exists_germ_injective, AlgebraicGeometry.Scheme.coprodPresheafObjIso_hom_snd_assoc, AlgebraicGeometry.Scheme.Hom.toImage_app_injective, PresheafOfModules.Derivation.postcomp_d_apply, AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv_appTop_coord, PresheafOfModules.Derivation.d_one, PresheafOfModules.Derivation.d_mul, AlgebraicGeometry.RingedSpace.mem_top_basicOpen, AlgebraicGeometry.stalkToFiberRingHom_homogeneousLocalizationToStalk, AlgebraicGeometry.Scheme.Hom.germ_stalkMap_apply, AlgebraicGeometry.Scheme.preimage_basicOpen_top, AlgebraicGeometry.IsAffineOpen.basicOpen_fromSpec_app, AlgebraicGeometry.Scheme.germToFunctionField_injective, instIsRightAdjointCommMonCatForget₂RingHomCarrierMonoidHomCarrier, AlgebraicGeometry.Scheme.Opens.ι_image_basicOpen_topIso_inv, PresheafOfModules.Monoidal.tensorObj_map_tmul, AlgebraicGeometry.Scheme.Hom.app_surjective, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.invApp_app_apply, AlgebraicGeometry.Scheme.Γevaluation_naturality_apply, AlgebraicGeometry.RingedSpace.res_zero, AlgebraicGeometry.Scheme.Hom.preimage_basicOpen_top, forget_map, TopCat.Sheaf.objSupIsoProdEqLocus_hom_fst, smoothSheafCommRing.evalHom_germ, PresheafOfModules.Derivation.congr_d, AlgebraicGeometry.exists_appTop_π_eq_of_isAffine_of_isLimit, AlgebraicGeometry.AffineSpace.SpecIso_inv_appTop_coord, smoothSheafCommRing.forgetStalk_hom_comp_evalHom_assoc, AlgebraicGeometry.Scheme.evaluation_eq_zero_iff_notMem_basicOpen, AlgebraicGeometry.LocallyRingedSpace.restrictStalkIso_inv_eq_germ_apply, AlgebraicGeometry.LocallyRingedSpace.stalkMap_inv_hom_apply, AlgebraicGeometry.Scheme.stalkSpecializes_stalkMap_apply, hom_inv_apply, AlgebraicGeometry.Scheme.AffineZariskiSite.coequifibered_iff_forall_isLocalizationAway, AlgebraicGeometry.AffineSpace.comp_homOfVector, AlgebraicGeometry.germ_injective_of_isIntegral, AlgebraicGeometry.isClosedImmersion_iff_isAffineHom, AlgebraicGeometry.AffineSpace.hom_ext_iff, forget₂CommSemiRing_preservesLimits, AlgebraicGeometry.Scheme.Hom.preimage_basicOpen, AlgEquiv.toUnder_inv_right_apply, AlgebraicGeometry.Scheme.basicOpen_res_eq, AlgebraicGeometry.Scheme.Hom.stalkFunctor_toImage_injective, AlgebraicGeometry.Scheme.image_basicOpen, AlgebraicGeometry.SurjectiveOnStalks.surj_on_stalks, commMon_forget₂_obj_coe, toAlgHom_apply, AlgebraicGeometry.SurjectiveOnStalks.stalkMap_surjective, AlgebraicGeometry.LocallyRingedSpace.restrictStalkIso_hom_eq_germ_apply, smoothSheafCommRing.ι_forgetStalk_inv_apply, forgetToRingCat_obj, Colimits.cocone_naturality_components, AlgebraicGeometry.Scheme.evaluation_naturality_apply, AlgebraicGeometry.Scheme.residue_surjective, smoothSheafCommRing.forgetStalk_hom_comp_evalHom, FilteredColimits.forget_preservesFilteredColimits, AlgebraicGeometry.LocallyRingedSpace.Γevaluation_naturality_apply, AlgebraicGeometry.StructureSheaf.toStalk_stalkSpecializes_apply, AlgebraicGeometry.Spec_zeroLocus_eq_zeroLocus, AlgebraicGeometry.Scheme.coprodPresheafObjIso_hom_snd, RingHom.surjective_iff_epi_and_finite, AlgebraicGeometry.Scheme.map_basicOpen, PresheafOfModules.Derivation'.d_app, TopCat.Sheaf.objSupIsoProdEqLocus_inv_snd, smoothSheafCommRing.ι_forgetStalk_inv_assoc, ModuleCat.Derivation.d_map, ofHom_apply, AlgebraicGeometry.IsClosedImmersion.hasAffineProperty, AlgHom.toUnder_right, PresheafOfModules.DifferentialsConstruction.relativeDifferentials'_map_d, AlgEquiv.toUnder_hom_right_apply, AlgebraicGeometry.IsAffineOpen.fromSpec_app_self_apply, AlgebraicGeometry.Scheme.stalkMap_hom_inv_apply, AlgebraicGeometry.Scheme.basicOpen_appLE, AlgebraicGeometry.Scheme.Hom.stalkSpecializes_stalkMap_apply, forget_map_apply, AlgebraicGeometry.AffineSpace.toSpecMvPolyIntEquiv_apply, forget₂Ring_preservesLimits, AlgebraicGeometry.LocallyRingedSpace.stalkMap_germ_apply, equalizer_ι_isLocalHom', FilteredColimits.forget₂Ring_preservesFilteredColimits, AlgebraicGeometry.stalkMap_toStalk_apply, AlgebraicGeometry.Scheme.stalkMap_germ_apply, RingHom.surjective_of_epi_of_finite, AlgebraicGeometry.Scheme.coprodPresheafObjIso_hom_fst, PresheafOfModules.DifferentialsConstruction.relativeDifferentials'_obj, TopCat.Presheaf.restrictOpenCommRingCat_apply, PresheafOfModules.Derivation.d_app, AlgebraicGeometry.Scheme.AffineZariskiSite.opensRange_relativeGluingData_map, AlgebraicGeometry.RingedSpace.basicOpen_res_eq, AlgebraicGeometry.Spec_zeroLocus, PresheafOfModules.Monoidal.tensorHom_app, PresheafOfModules.DifferentialsConstruction.relativeDifferentials'_map, inv_hom_apply, PresheafOfModules.Derivation'.app_apply, comp_apply, AlgebraicGeometry.StructureSheaf.instIsLocalizedModuleCarrierStalkAbPresheafOpensCarrierTopModuleStructurePresheafPrimeComplAsIdealToStalkₗ, AlgebraicGeometry.basicOpen_eq_of_affine', AlgebraicGeometry.LocallyRingedSpace.stalkSpecializes_stalkMap_apply, TopCat.Sheaf.objSupIsoProdEqLocus_hom_snd, AlgebraicGeometry.stalkToFiberRingHom_germ, TopCat.Sheaf.objSupIsoProdEqLocus_inv_eq_iff, AlgebraicGeometry.exists_eq_pow_mul_of_isAffineOpen, AlgebraicGeometry.specTargetImageFactorization_app_injective, AlgebraicGeometry.Scheme.IsGermInjectiveAt.cond, AlgebraicGeometry.IsOpenImmersion.app_ΓIso_hom_apply, AlgebraicGeometry.StructureSheaf.stalkAlgebra_map, equalizer_ι_isLocalHom, AlgebraicGeometry.Scheme.stalkMap_inv_hom_apply, AlgebraicGeometry.AffineSpace.reindex_appTop_coord, AlgebraicGeometry.Scheme.Hom.appLE_appIso_inv_apply, AlgebraicGeometry.RingedSpace.isUnit_res_basicOpen, AlgebraicGeometry.exists_appTop_π_eq_of_isLimit, AlgebraicGeometry.Proj.stalkIso'_germ, AlgebraicGeometry.Scheme.mem_basicOpen_top, AlgebraicGeometry.Scheme.Opens.ι_image_basicOpen', AlgebraicGeometry.LocallyRingedSpace.preimage_basicOpen, AlgebraicGeometry.AffineSpace.homOverEquiv_apply, AlgebraicGeometry.Scheme.basicOpen_res, TopCat.Sheaf.objSupIsoProdEqLocus_inv_fst, AlgebraicGeometry.Scheme.Hom.stalkMap_hom_inv_apply, AlgebraicGeometry.IsClosedImmersion.isAffine_surjective_of_isAffine, AlgebraicGeometry.germ_eq_zero_of_pow_mul_eq_zero, AlgebraicGeometry.AffineSpace.toSpecMvPolyIntEquiv_comp, AlgebraicGeometry.LocallyRingedSpace.Γevaluation_eq_zero_iff_notMem_basicOpen, AlgebraicGeometry.Scheme.IdealSheafData.subschemeι_app_surjective, AlgebraicGeometry.Scheme.map_basicOpen_map, AlgebraicGeometry.Scheme.AffineZariskiSite.PreservesLocalization.opensRange_map, AlgebraicGeometry.Scheme.mem_basicOpen'', AlgebraicGeometry.specTargetImageRingHom_surjective, AlgebraicGeometry.basicOpen_eq_of_affine, AlgebraicGeometry.LocallyRingedSpace.isUnit_res_toΓSpecMapBasicOpen, commMon_forget₂_map, AlgebraicGeometry.Scheme.Hom.appIso_inv_app_apply, AlgebraicGeometry.IsAffineOpen.fromSpec_preimage_basicOpen', AlgebraicGeometry.Scheme.basicOpen_restrict, AlgebraicGeometry.IsAffineOpen.app_basicOpen_eq_away_map, forget_preservesLimits, AlgebraicGeometry.Scheme.preimage_basicOpen, AlgebraicGeometry.Scheme.exists_le_and_germ_injective, AlgebraicGeometry.AffineSpace.comp_homOfVector_assoc, forgetToRingCat_map_hom, AlgebraicGeometry.Scheme.Hom.stalkMap_inv_hom_apply, AlgebraicGeometry.Scheme.basicOpen_eq_bot_iff_forall_evaluation_eq_zero, AlgebraicGeometry.LocallyRingedSpace.evaluation_eq_zero_iff_notMem_basicOpen, AlgebraicGeometry.Proj.res_apply, PresheafOfModules.germ_smul, AlgebraicGeometry.AffineSpace.map_appTop_coord, smoothSheafCommRing.forgetStalk_hom_comp_evalHom_apply, AlgebraicGeometry.LocallyRingedSpace.stalkMap_hom_inv_apply, AlgebraicGeometry.LocallyRingedSpace.evaluation_naturality_apply, AlgebraicGeometry.map_injective_of_isIntegral, AlgebraicGeometry.exists_eq_pow_mul_of_isCompact_of_isQuasiSeparated, AlgebraicGeometry.AffineSpace.homOfVector_appTop_coord, AlgebraicGeometry.Proj.stalkIso'_symm_mk, CommAlgCat.forget₂_commRingCat_map, AlgebraicGeometry.RingedSpace.mem_basicOpen, AlgebraicGeometry.Scheme.Hom.app_injective, AlgebraicGeometry.StructureSheaf.algebraMap_germ_apply, AlgebraicGeometry.Scheme.localRingHom_comp_stalkIso_apply, instIsRightAdjointForgetRingHomCarrier, AlgebraicGeometry.IsAffineOpen.appLE_eq_away_map, AlgebraicGeometry.Scheme.Modules.map_smul, AlgebraicGeometry.RingedSpace.mem_basicOpen', AlgebraicGeometry.Scheme.mem_basicOpen, AlgebraicGeometry.IsOpenImmersion.map_ΓIso_inv_apply, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.ofRestrict_invApp_apply, AlgebraicGeometry.homogeneousLocalizationToStalk_stalkToFiberRingHom, AlgebraicGeometry.Scheme.Hom.stalkMap_surjective, AlgebraicGeometry.IsAffineOpen.mem_ideal_iff
instHasForget₂RingHomCarrierCommMonCatMonoidHomCarrier 📖	CompOp	3 math math: instIsRightAdjointCommMonCatForget₂RingHomCarrierMonoidHomCarrier, commMon_forget₂_obj_coe, commMon_forget₂_map
instInhabited 📖	CompOp	—
ofHom 📖	CompOp	164 math math: tensorProd_map_right, AlgebraicGeometry.AffineSpace.map_Spec_map, AlgebraicGeometry.IsPreimmersion.of_isLocalization, AlgebraicGeometry.Proj.awayMap_awayToSection_assoc, AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction_counit_app, AlgebraicGeometry.StructureSheaf.globalSectionsIso_inv, AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv, Localization.epi', pushoutCocone_pt, AlgebraicGeometry.ProjectiveSpectrum.Proj.toStalk_specStalkEquiv, AlgebraicGeometry.Proj.fromOfGlobalSections_toSpecZero_assoc, AlgebraicGeometry.AffineSpace.toSpecMvPolyIntEquiv_symm_apply, AlgebraicGeometry.AffineSpace.SpecIso_hom_appTop, AlgebraicGeometry.diagonal_SpecMap, AlgebraicGeometry.ι_sigmaSpec, AlgebraicGeometry.coprodSpec_inr, Algebra.FormallyUnramified.isOpenImmersion_SpecMap_lmul, AlgebraicGeometry.Scheme.basic_open_isOpenImmersion, AlgebraicGeometry.StructureSheaf.toOpen_comp_comap, coyoneda_map_app, ofHom_id, AlgebraicGeometry.Scheme.Hom.toNormalization_app_preimage, AlgebraicGeometry.StructureSheaf.to_basicOpen_epi, AlgebraicGeometry.Scheme.isOpenImmersion_SpecMap_localizationAway, AlgebraicGeometry.Scheme.instIsOpenImmersionMapOfHomAwayAlgebraMap, pushoutCocone_inl, coyoneda_obj_map, ofHom_hom, AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv_over_assoc, AlgebraicGeometry.ProjectiveSpectrum.Proj.awayToΓ_ΓToStalk, pushout_inr_tensorProdObjIsoPushoutObj_inv_right_assoc, AlgebraicGeometry.Scheme.Hom.ι_toNormalization, AlgebraicGeometry.Proj.pullbackAwayιIso_hom_SpecMap_awayMap_right_assoc, AlgebraicGeometry.Scheme.Hom.liftQuotient_comp, AlgebraicGeometry.Proj.pullbackAwayιIso_inv_snd_assoc, AlgebraicGeometry.StructureSheaf.toPushforwardStalkAlgHom_apply, AlgebraicGeometry.Scheme.ΓSpecIso_inv, pushoutCocone_inr, Localization.epi, AlgebraicGeometry.Proj.SpecMap_awayMap_awayι_assoc, AlgebraicGeometry.pullbackSpecIso_inv_snd, AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction_counit_app', AlgebraicGeometry.Proj.pullbackAwayιIso_inv_fst, AlgebraicGeometry.Proj.basicOpenToSpec_SpecMap_awayMap, coproductCocone_inl, AlgebraicGeometry.pullbackSpecIso_hom_base, AlgebraicGeometry.Spec_map_localization_isIso, AlgebraicGeometry.StructureSheaf.isLocalizedModule_toPushforwardStalkAlgHom_aux, AlgebraicGeometry.StructureSheaf.algebraMap_self_map, isPushout_tensorProduct, mkUnder_hom, AlgebraicGeometry.Spec.map_appLE, RingEquiv.toCommRingCatIso_inv, coproductCocone_inr, AlgebraicGeometry.Proj.pullbackAwayιIso_inv_snd, AlgebraicGeometry.Proj.pullbackAwayιIso_inv_fst_assoc, AlgebraicGeometry.pullbackSpecIso_inv_fst'_assoc, AlgebraicGeometry.Spec.germ_stalkMapIso_hom_assoc, isPushout_of_isPushout, AlgebraicGeometry.LocallyRingedSpace.residue_comp_residueFieldMap_eq_stalkMap_comp_residue, AlgebraicGeometry.StructureSheaf.toOpen_comp_comap_assoc, AlgebraicGeometry.Proj.pullbackAwayιIso_hom_SpecMap_awayMap_right, AlgebraicGeometry.ΓSpec.toOpen_comp_locallyRingedSpaceAdjunction_homEquiv_app, RingEquiv.toCommRingCatIso_hom, AlgebraicGeometry.StructureSheaf.globalSectionsIso_hom, AlgebraicGeometry.Scheme.Spec.residue_residueFieldIso_hom_assoc, AlgebraicGeometry.IsClosedImmersion.spec_of_quotient_mk, epi_iff_tmul_eq_tmul, AlgebraicGeometry.ΓSpec.unop_locallyRingedSpaceAdjunction_counit_app', AlgebraicGeometry.Scheme.IdealSheafData.subschemeι_app, AlgebraicGeometry.StructureSheaf.isLocalizedModule_toPushforwardStalkAlgHom, AlgebraicGeometry.Proj.pullbackAwayιIso_hom_SpecMap_awayMap_left, AlgebraicGeometry.pullbackSpecIso_inv_fst, pushout_inl_tensorProdObjIsoPushoutObj_inv_right_assoc, epi_iff_epi, AlgebraicGeometry.Proj.awayι_toSpecZero, monoidAlgebra_map, AlgebraicGeometry.Spec.sheafedSpaceMap_hom_c_app, IsLocalization.epi, AlgebraicGeometry.Proj.awayι_toSpecZero_assoc, AlgebraicGeometry.ValuativeCommSq.commSq, isPushout_iff_isPushout, pushout_inl_tensorProdObjIsoPushoutObj_inv_right, AlgebraicGeometry.Proj.pullbackAwayιIso_hom_SpecMap_awayMap_left_assoc, AlgebraicGeometry.AffineSpace.map_SpecMap, ofHom_apply, AlgebraicGeometry.Proj.fromOfGlobalSections_toSpecZero, AlgebraicGeometry.Scheme.Spec.algebraMap_residueFieldIso_inv, AlgebraicGeometry.AffineSpace.SpecIso_inv_over_assoc, AlgebraicGeometry.pullbackSpecIso_inv_snd_assoc, AlgebraicGeometry.coprodSpec_inr_assoc, AlgebraicGeometry.IsClosedImmersion.Spec_iff, AlgebraicGeometry.Spec.algebraMap_stalkIso_inv, AlgebraicGeometry.pullbackSpecIso_inv_fst_assoc, AlgebraicGeometry.Scheme.Hom.ι_toNormalization_assoc, AlgebraicGeometry.Scheme.Hom.fromNormalization_app_assoc, AlgebraicGeometry.HasRingHomProperty.of_isLocalAtSource_of_isLocalAtTarget, hom_ofHom, localization_unit_isIso, AlgebraicGeometry.Scheme.toOpen_eq, AlgebraicGeometry.instIsOpenImmersionMapOfHomForallEvalRingHom, AlgebraicGeometry.StructureSheaf.algebraMap_germ_assoc, TopCat.Presheaf.SubmonoidPresheaf.toLocalizationPresheaf_app, AlgebraicGeometry.Proj.basicOpenToSpec_SpecMap_awayMap_assoc, coproductCoconeIsColimit_desc, AlgebraicGeometry.pullbackSpecIso_hom_fst, AlgebraicGeometry.HasRingHomProperty.isLocal_ringHomProperty_of_isZariskiLocalAtSource_of_isZariskiLocalAtTarget, tensorProdIsoPushout_app, AlgebraicGeometry.StructureSheaf.algebraMap_germ, AlgebraicGeometry.AffineSpace.isoOfIsAffine_hom_appTop, AlgebraicGeometry.coprodSpec_inl_assoc, AlgebraicGeometry.Proj.awayMap_awayToSection, AlgebraicGeometry.Proj.SpecMap_awayMap_awayι, AlgebraicGeometry.basicOpenIsoSpecAway_inv_homOfLE_assoc, AlgebraicGeometry.isIso_SpecMap_stakMap_localization, AlgebraicGeometry.ι_sigmaSpec_assoc, AlgebraicGeometry.AffineSpace.isoOfIsAffine_hom, AlgebraicGeometry.basicOpenIsoSpecAway_inv_homOfLE, AlgebraicGeometry.coprodSpec_inl, AlgebraicGeometry.Spec.algebraMap_stalkIso_inv_assoc, AlgebraicGeometry.ΓSpec.toSpecΓ_of, AlgebraicGeometry.StructureSheaf.toOpen_germ, AlgebraicGeometry.pullbackSpecIso_hom_base_assoc, HomTopology.mvPolynomialHomeomorph_apply_fst, AlgebraicGeometry.Scheme.affineOpenCoverOfSpanRangeEqTop_f, AlgebraicGeometry.Scheme.Spec.algebraMap_residueFieldIso_inv_assoc, AlgebraicGeometry.ProjectiveSpectrum.Proj.awayToSection_germ, AlgebraicGeometry.pullbackSpecIso_hom_fst'_assoc, AlgebraicGeometry.geometrically_iff_of_commRing_of_isClosedUnderIsomorphisms, AlgebraicGeometry.pullbackSpecIso_inv_fst', monoidAlgebra_obj, AlgebraicGeometry.diagonal_Spec_map, AlgebraicGeometry.AffineSpace.SpecIso_inv_over, AlgebraicGeometry.IsOpenImmersion.of_isLocalization, AlgebraicGeometry.Spec.germ_stalkMapIso_hom, localization_unit_isIso', AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv_over, ofHom_comp, AlgebraicGeometry.IsAffineOpen.app_basicOpen_eq_away_map, AlgebraicGeometry.pullbackSpecIso_hom_fst', AlgebraicGeometry.Scheme.IdealSheafData.glueDataObjι_ι, AlgebraicGeometry.Scheme.Hom.toImage_app, AlgebraicGeometry.LocallyRingedSpace.toΓSpecSheafedSpace_app_spec, AlgebraicGeometry.StructureSheaf.toOpen_res, AlgebraicGeometry.LocallyRingedSpace.toΓSpecSheafedSpace_app_spec_assoc, AlgebraicGeometry.LocallyRingedSpace.toΓSpecCApp_spec, AlgebraicGeometry.pullbackSpecIso_hom_snd, AlgebraicGeometry.Scheme.Hom.normalizationObjIso_hom_val, CommAlgCat.forget₂_commRingCat_map, AlgebraicGeometry.localRingHom_comp_stalkIso, AlgebraicGeometry.Scheme.localRingHom_comp_stalkIso, AlgebraicGeometry.Scheme.Spec.residue_residueFieldIso_hom, AlgebraicGeometry.Scheme.Hom.liftQuotient_comp_assoc, coproductCocone_ι, AlgebraicGeometry.pullbackSpecIso_hom_fst_assoc, AlgebraicGeometry.Scheme.Hom.fromNormalization_app, AlgebraicGeometry.HasRingHomProperty.of_isZariskiLocalAtSource_of_isZariskiLocalAtTarget, AlgebraicGeometry.ΓSpec.toSpecΓ_unop, AlgebraicGeometry.IsAffineOpen.appLE_eq_away_map, isPushout_of_isLocalization, AlgebraicGeometry.LocallyRingedSpace.toΓSpecCApp_iff, AlgebraicGeometry.Spec.map_app, pushout_inr_tensorProdObjIsoPushoutObj_inv_right, AlgebraicGeometry.pullbackSpecIso_hom_snd_assoc

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
coe_of 📖	mathematical	—	carrier of	—	—
commMon_forget₂_map 📖	mathematical	—	CategoryTheory.Functor.map CommRingCat instCategory CommMonCat CommMonCat.instCategory CategoryTheory.HasForget₂.forget₂ RingHom carrier Semiring.toNonAssocSemiring CommSemiring.toSemiring CommRing.toCommSemiring commRing RingHom.instFunLike instConcreteCategoryRingHomCarrier MonoidHom CommMonCat.carrier MulOneClass.toMulOne Monoid.toMulOneClass CommMonoid.toMonoid CommMonCat.str MonoidHom.instFunLike CommMonCat.instConcreteCategoryMonoidHomCarrier instHasForget₂RingHomCarrierCommMonCatMonoidHomCarrier CommMonCat.ofHom CommRing.toCommMonoid MonoidHomClass.toMonoidHom Hom.hom	—	—
commMon_forget₂_obj_coe 📖	mathematical	—	CommMonCat.carrier CategoryTheory.Functor.obj CommRingCat instCategory CommMonCat CommMonCat.instCategory CategoryTheory.HasForget₂.forget₂ RingHom carrier Semiring.toNonAssocSemiring CommSemiring.toSemiring CommRing.toCommSemiring commRing RingHom.instFunLike instConcreteCategoryRingHomCarrier MonoidHom MulOneClass.toMulOne Monoid.toMulOneClass CommMonoid.toMonoid CommMonCat.str MonoidHom.instFunLike CommMonCat.instConcreteCategoryMonoidHomCarrier instHasForget₂RingHomCarrierCommMonCatMonoidHomCarrier	—	—
comp_apply 📖	mathematical	—	DFunLike.coe carrier CategoryTheory.ConcreteCategory.hom CommRingCat instCategory RingHom Semiring.toNonAssocSemiring CommSemiring.toSemiring CommRing.toCommSemiring commRing RingHom.instFunLike instConcreteCategoryRingHomCarrier CategoryTheory.CategoryStruct.comp CategoryTheory.Category.toCategoryStruct	—	—
forgetReflectIsos 📖	mathematical	—	CategoryTheory.Functor.ReflectsIsomorphisms CommRingCat instCategory CategoryTheory.types CategoryTheory.forget RingHom carrier Semiring.toNonAssocSemiring CommSemiring.toSemiring CommRing.toCommSemiring commRing RingHom.instFunLike instConcreteCategoryRingHomCarrier	—	Equiv.left_inv Equiv.right_inv MonoidHom.map_mul' RingHom.map_add' CategoryTheory.Iso.isIso_hom
forgetToRingCat_map_hom 📖	mathematical	—	RingCat.Hom.hom CategoryTheory.Functor.obj CommRingCat instCategory RingCat RingCat.instCategory CategoryTheory.forget₂ RingHom carrier Semiring.toNonAssocSemiring CommSemiring.toSemiring CommRing.toCommSemiring commRing RingHom.instFunLike instConcreteCategoryRingHomCarrier RingCat.carrier Ring.toSemiring RingCat.ring RingCat.instConcreteCategoryRingHomCarrier hasForgetToRingCat CategoryTheory.Functor.map Hom.hom	—	—
forgetToRingCat_obj 📖	mathematical	—	RingCat.carrier CategoryTheory.Functor.obj CommRingCat instCategory RingCat RingCat.instCategory CategoryTheory.forget₂ RingHom carrier Semiring.toNonAssocSemiring CommSemiring.toSemiring CommRing.toCommSemiring commRing RingHom.instFunLike instConcreteCategoryRingHomCarrier Ring.toSemiring RingCat.ring RingCat.instConcreteCategoryRingHomCarrier hasForgetToRingCat	—	—
forget_map 📖	mathematical	—	CategoryTheory.Functor.map CommRingCat instCategory CategoryTheory.types CategoryTheory.forget RingHom carrier Semiring.toNonAssocSemiring CommSemiring.toSemiring CommRing.toCommSemiring commRing RingHom.instFunLike instConcreteCategoryRingHomCarrier DFunLike.coe CategoryTheory.ConcreteCategory.hom	—	—
forget_map_apply 📖	mathematical	—	CategoryTheory.Functor.map CommRingCat instCategory CategoryTheory.types CategoryTheory.forget RingHom carrier Semiring.toNonAssocSemiring CommSemiring.toSemiring CommRing.toCommSemiring commRing RingHom.instFunLike instConcreteCategoryRingHomCarrier DFunLike.coe CategoryTheory.ConcreteCategory.hom	—	—
forget_obj 📖	mathematical	—	CategoryTheory.Functor.obj CommRingCat instCategory CategoryTheory.types CategoryTheory.forget RingHom carrier Semiring.toNonAssocSemiring CommSemiring.toSemiring CommRing.toCommSemiring commRing RingHom.instFunLike instConcreteCategoryRingHomCarrier	—	—
hom_comp 📖	mathematical	—	Hom.hom CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct instCategory RingHom.comp carrier Semiring.toNonAssocSemiring CommSemiring.toSemiring CommRing.toCommSemiring commRing	—	—
hom_ext 📖	—	Hom.hom	—	—	Hom.ext
hom_ext_iff 📖	mathematical	—	Hom.hom	—	hom_ext
hom_id 📖	mathematical	—	Hom.hom CategoryTheory.CategoryStruct.id CommRingCat CategoryTheory.Category.toCategoryStruct instCategory RingHom.id carrier Semiring.toNonAssocSemiring CommSemiring.toSemiring CommRing.toCommSemiring commRing	—	—
hom_inv_apply 📖	mathematical	—	DFunLike.coe carrier CategoryTheory.ConcreteCategory.hom CommRingCat instCategory RingHom Semiring.toNonAssocSemiring CommSemiring.toSemiring CommRing.toCommSemiring commRing RingHom.instFunLike instConcreteCategoryRingHomCarrier CategoryTheory.Iso.hom CategoryTheory.Iso.inv	—	CategoryTheory.Iso.inv_hom_id_apply
hom_ofHom 📖	mathematical	—	Hom.hom of ofHom	—	—
id_apply 📖	mathematical	—	DFunLike.coe carrier CategoryTheory.ConcreteCategory.hom CommRingCat instCategory RingHom Semiring.toNonAssocSemiring CommSemiring.toSemiring CommRing.toCommSemiring commRing RingHom.instFunLike instConcreteCategoryRingHomCarrier CategoryTheory.CategoryStruct.id CategoryTheory.Category.toCategoryStruct	—	—
instFullRingCatForget₂RingHomCarrierCarrier 📖	mathematical	—	CategoryTheory.Functor.Full CommRingCat instCategory RingCat RingCat.instCategory CategoryTheory.forget₂ RingHom carrier Semiring.toNonAssocSemiring CommSemiring.toSemiring CommRing.toCommSemiring commRing RingHom.instFunLike instConcreteCategoryRingHomCarrier RingCat.carrier Ring.toSemiring RingCat.ring RingCat.instConcreteCategoryRingHomCarrier hasForgetToRingCat	—	CategoryTheory.Functor.FullyFaithful.full
inv_hom_apply 📖	mathematical	—	DFunLike.coe carrier CategoryTheory.ConcreteCategory.hom CommRingCat instCategory RingHom Semiring.toNonAssocSemiring CommSemiring.toSemiring CommRing.toCommSemiring commRing RingHom.instFunLike instConcreteCategoryRingHomCarrier CategoryTheory.Iso.inv CategoryTheory.Iso.hom	—	CategoryTheory.Iso.hom_inv_id_apply
ofHom_apply 📖	mathematical	—	DFunLike.coe of carrier CategoryTheory.ConcreteCategory.hom CommRingCat instCategory RingHom Semiring.toNonAssocSemiring CommSemiring.toSemiring CommRing.toCommSemiring commRing RingHom.instFunLike instConcreteCategoryRingHomCarrier ofHom	—	—
ofHom_comp 📖	mathematical	—	ofHom RingHom.comp Semiring.toNonAssocSemiring CommSemiring.toSemiring CommRing.toCommSemiring CategoryTheory.CategoryStruct.comp CommRingCat CategoryTheory.Category.toCategoryStruct instCategory of	—	—
ofHom_hom 📖	mathematical	—	ofHom carrier commRing Hom.hom	—	—
ofHom_id 📖	mathematical	—	ofHom RingHom.id Semiring.toNonAssocSemiring CommSemiring.toSemiring CommRing.toCommSemiring CategoryTheory.CategoryStruct.id CommRingCat CategoryTheory.Category.toCategoryStruct instCategory of	—	—
of_carrier 📖	mathematical	—	of carrier commRing	—	—

CommRingCat.Hom

Definitions

Name	Category	Theorems
hom 📖	CompOp	201 math math: AlgebraicGeometry.AffineSpace.map_Spec_map, CommRingCat.KaehlerDifferential.map_d, AlgebraicGeometry.IsAffineOpen.isLocalization_of_eq_basicOpen, AlgebraicGeometry.Etale.etale_appLE, AlgebraicGeometry.LocallyQuasiFinite.quasiFinite_appLE, AlgebraicGeometry.LocallyOfFiniteType.finiteType_of_affine_subset, AlgebraicGeometry.StructureSheaf.comap_id, AlgebraicGeometry.AffineSpace.SpecIso_hom_appTop, AlgebraicGeometry.isIso_SpecMap_iff, AlgebraicGeometry.HasRingHomProperty.stalkMap, AlgebraicGeometry.HasAffineProperty.SpecMap_iff_of_affineAnd, AlgebraicGeometry.Scheme.ideal_ker_le_ker_ΓSpecIso_inv_comp, CommRingCat.moduleCatRestrictScalarsPseudofunctor_mapComp, AlgebraicGeometry.localRingHom_comp_stalkIso_apply, AlgebraicGeometry.SmoothOfRelativeDimension.exists_isStandardSmoothOfRelativeDimension, AlgebraicGeometry.Scheme.IdealSheafData.ideal_le_comap_ideal, AlgebraicGeometry.ValuativeCriterion.Existence.instIsLocalHomCarrierRingHomHomHomCommRingCat, AlgebraicGeometry.locallyQuasiFinite_iff, AlgebraicGeometry.affineLocally_iff_forall_isAffineOpen, AlgebraicGeometry.Scheme.Hom.formallyUnramified_appLE, CommRingCat.coyoneda_map_app, AlgebraicGeometry.Scheme.Hom.toNormalization_app_preimage, AlgebraicGeometry.Scheme.IdealSheafData.ideal_le_ker_glueDataObjι, AlgebraicGeometry.SpecToEquivOfLocalRing_apply_snd_coe, AlgebraicGeometry.SurjectiveOnStalks.iff_of_isAffine, AlgebraicGeometry.stalkwise_SpecMap_iff, CommRingCat.coyoneda_obj_map, AlgebraicGeometry.IsIntegralHom.isIntegral_app, AlgebraicGeometry.Scheme.ker_ideal_of_isPullback_of_isOpenImmersion, CommRingCat.ofHom_hom, AlgebraicGeometry.Scheme.IdealSheafData.ideal_map, AlgebraicGeometry.Scheme.Hom.ι_toNormalization, AlgebraicGeometry.Scheme.Hom.smooth_appLE, AlgebraicGeometry.IsAffineOpen.ideal_le_iff, AlgebraicGeometry.Scheme.Hom.QuasiFiniteAt.quasiFiniteAt, RingHom.RespectsIso.cancel_right_isIso, PresheafOfModules.Monoidal.tensorObj_map_tmul, TopCat.Presheaf.stalk_open_algebraMap, smoothSheafCommRing.forgetStalk_inv_comp_eval_assoc, AlgebraicGeometry.affineLocally_iff_affineOpens_le, AlgebraicGeometry.HasRingHomProperty.iff_of_iSup_eq_top, AlgebraicGeometry.IsPreimmersion.Spec_map_iff, smoothSheafCommRing.forgetStalk_inv_comp_eval, AlgebraicGeometry.Scheme.ker_of_isAffine, AlgebraicGeometry.IsAffineOpen.algebraMap_Spec_obj, AlgebraicGeometry.LocallyRingedSpace.isLocalHomStalkMap, AlgebraicGeometry.locallyOfFiniteType_iff, AlgebraicGeometry.Flat.flat_and_surjective_iff_faithfullyFlat_of_isAffine, AlgebraicGeometry.LocallyRingedSpace.coe_toΓSpecSheafedSpace_hom_base_hom_apply_asIdeal, smoothSheafCommRing.forgetStalk_inv_comp_eval_apply, AlgebraicGeometry.isIntegral_appTop_of_universallyClosed, AlgebraicGeometry.LocallyOfFiniteType.finiteType_appLE, AlgebraicGeometry.Spec.map_appLE, AlgebraicGeometry.Scheme.Hom.finitePresentation_appLE, CommRingCat.instIsLocalHomCarrierObjWalkingParallelPairFunctorConstPtEqualizerForkZeroParallelPairRingHomHomι, CommRingCat.moduleCatRestrictScalarsPseudofunctor_map, AlgebraicGeometry.FormallyUnramified.formallyUnramified_of_affine_subset, AlgebraicGeometry.formallySmooth_stalkMap_iff, TopCat.Sheaf.objSupIsoProdEqLocus_hom_fst, AlgebraicGeometry.HasRingHomProperty.iff_appLE, AlgebraicGeometry.HasRingHomProperty.appLE, AlgebraicGeometry.Scheme.Hom.ker_apply, AlgebraicGeometry.Scheme.IdealSheafData.ideal_comap_of_isOpenImmersion, CommRingCat.hom_comp, CommRingCat.HomTopology.isClosedEmbedding_hom, AlgebraicGeometry.Scheme.IdealSheafData.map_ideal, AlgebraicGeometry.Scheme.zeroLocus_map, AlgebraicGeometry.Scheme.Hom.finite_app, AlgebraicGeometry.Scheme.AffineZariskiSite.coequifibered_iff_forall_isLocalizationAway, AlgebraicGeometry.FormallyUnramified.instIsSeparableCarrierResidueFieldCoeContinuousMapCarrierCarrierCommRingCatHomTopCatBaseOfLocallyOfFiniteType, AlgebraicGeometry.Scheme.Hom.ideal_ker_le, AlgebraicGeometry.smooth_iff, AlgebraicGeometry.stalkMap_toStalk, CommRingCat.HomTopology.mvPolynomialHomeomorph_symm_apply_hom, AlgebraicGeometry.LocallyRingedSpace.Hom.prop, TopCat.Presheaf.SubmonoidPresheaf.map, AlgebraicGeometry.HasRingHomProperty.appTop, CommRingCat.HomTopology.isEmbedding_hom, CommRingCat.moduleCatExtendScalarsPseudofunctor_map, AlgebraicGeometry.IsFinite.instHasAffinePropertyAndIsAffineFiniteCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTopHomAppTop, AlgebraicGeometry.SpecToEquivOfLocalRing_apply_fst, AlgebraicGeometry.Spec.map_apply, AlgebraicGeometry.Spec.sheafedSpaceMap_hom_c_app, AlgebraicGeometry.targetAffineLocally_affineAnd_iff, AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_stalk_isLocalHom, CommRingCat.hom_ext_iff, RingHom.surjective_iff_epi_and_finite, TopCat.Sheaf.objSupIsoProdEqLocus_inv_snd, AlgebraicGeometry.exists_smooth_of_formallySmooth_stalk, AlgebraicGeometry.LocallyOfFiniteType.stalkMap, AlgebraicGeometry.Scheme.Hom.etale_appLE, AlgebraicGeometry.AffineSpace.map_SpecMap, AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalHom, AlgebraicGeometry.Scheme.IdealSheafData.isLocalization_away, RingHom.RespectsIso.cancel_left_isIso, AlgebraicGeometry.IsAffineOpen.comap_primeIdealOf_appLE, CommRingCat.HomTopology.continuous_apply, AlgebraicGeometry.LocallyRingedSpace.isLocalHomStalkMap', AlgebraicGeometry.Scheme.IdealSheafData.ofIdealTop_ideal, AlgebraicGeometry.SurjectiveOnStalks.Spec_iff, isLocalHom_of_isIso, AlgebraicGeometry.Flat.flat_of_affine_subset, AlgebraicGeometry.stalkwise_Spec_map_iff, AlgebraicGeometry.HasAffineProperty.affineAnd_iff, AlgebraicGeometry.Scheme.IdealSheafData.ker_subschemeι_app, AlgebraicGeometry.Scheme.Hom.ι_toNormalization_assoc, CommRingCat.equalizer_ι_isLocalHom', AlgebraicGeometry.Scheme.isLocalHom_stalkClosedPointTo', AlgebraicGeometry.Scheme.Hom.fromNormalization_app_assoc, AlgebraicGeometry.stalkMap_toStalk_apply, CommRingCat.hom_ofHom, AlgebraicGeometry.Scheme.Hom.isIntegral_app, AlgebraicGeometry.IsFinite.SpecMap_iff, CommRingCat.coproductCoconeIsColimit_desc, AlgebraicGeometry.Flat.flat_appLE, AlgebraicGeometry.isIntegralHom_iff, AlgebraicGeometry.LocallyOfFinitePresentation.finitePresentation_appLE, AlgebraicGeometry.IsPreimmersion.SpecMap_iff, AlgebraicGeometry.Scheme.Hom.finiteType_appLE, AlgebraicGeometry.Flat.stalkMap, AlgebraicGeometry.AffineSpace.isoOfIsAffine_hom_appTop, AlgebraicGeometry.Spec.coe_toTop_map_hom_apply_asIdeal, smoothSheafCommRing.ι_forgetStalk_hom_apply, TopCat.Sheaf.objSupIsoProdEqLocus_hom_snd, TopCat.Sheaf.objSupIsoProdEqLocus_inv_eq_iff, AlgebraicGeometry.IsFinite.finite_app, smoothSheafCommRing.ι_forgetStalk_hom_assoc, AlgebraicGeometry.finite_appTop_of_universallyClosed, AlgebraicGeometry.IsIntegralHom.integral_app, CommRingCat.coyonedaUnique_inv_app_hom_apply, AlgebraicGeometry.FormallyUnramified.stalkMap, AlgebraicGeometry.Scheme.IdealSheafData.map_ideal', AlgebraicGeometry.StructureSheaf.comap_id_eq_map, AlgebraicGeometry.HasRingHomProperty.Spec_iff, AlgebraicGeometry.AffineSpace.isoOfIsAffine_hom, CommRingCat.equalizer_ι_isLocalHom, AlgebraicGeometry.affineAnd_apply, AlgebraicGeometry.HasRingHomProperty.iff_exists_appLE_locally, AlgebraicGeometry.targetAffineLocally_affineAnd_iff', AlgebraicGeometry.smoothOfRelativeDimension_iff, AlgebraicGeometry.HasRingHomProperty.iff_exists_appLE, AlgebraicGeometry.Scheme.Hom.mem_smoothLocus, AlgebraicGeometry.formallyUnramified_iff, CommRingCat.moduleCatExtendScalarsPseudofunctor_mapComp, CommRingCat.coyonedaUnique_hom_app_hom_apply, AlgebraicGeometry.IsIntegralHom.SpecMap_iff, TopCat.Sheaf.objSupIsoProdEqLocus_inv_fst, AlgebraicGeometry.Smooth.exists_isStandardSmooth, AlgebraicGeometry.Smooth.smooth_appLE, smoothSheafCommRing.ι_forgetStalk_hom, AlgebraicGeometry.Flat.iff_flat_stalkMap, AlgebraicGeometry.Scheme.preimage_zeroLocus, AlgebraicGeometry.Spec.map_base, AlgebraicGeometry.SpecToEquivOfLocalRing_symm_apply, AlgebraicGeometry.IsIntegralHom.hasAffineProperty, AlgebraicGeometry.LocallyOfFinitePresentation.finitePresentation_of_affine_subset, AlgebraicGeometry.isFinite_iff, AlgebraicGeometry.etale_iff, AlgebraicGeometry.flat_and_surjective_SpecMap_iff, CategoryTheory.Iso.commRingCatIsoToRingEquiv_toRingHom, CommRingCat.hom_id, isLocalHom_of_iso, AlgebraicGeometry.locallyOfFinitePresentation_iff, CommRingCat.commMon_forget₂_map, CommRingCat.isLocalHom_comp, AlgebraicGeometry.Scheme.IdealSheafData.ker_glueDataObjι_appTop, AlgebraicGeometry.FormallyUnramified.formallyUnramified_appLE, AlgebraicGeometry.HasRingHomProperty.stalkMap_of_respectsIso, AlgebraicGeometry.Scheme.isLocalHom_stalkClosedPointTo, AlgebraicGeometry.IsAffineOpen.app_basicOpen_eq_away_map, AlgebraicGeometry.SpecToEquivOfLocalRing_eq_iff, AlgebraicGeometry.Scheme.Hom.flat_appLE, CommRingCat.forgetToRingCat_map_hom, AlgebraicGeometry.Scheme.Hom.toImage_app, AlgebraicGeometry.exists_lift_of_germInjective_aux, AlgebraicGeometry.Scheme.image_zeroLocus, AlgebraicGeometry.HasRingHomProperty.iff_of_source_openCover, AlgebraicGeometry.LocallyRingedSpace.isLocalHom_stalkMap_congr, AlgebraicGeometry.Scheme.Hom.normalizationObjIso_hom_val, AlgebraicGeometry.Scheme.Hom.instIsLocalHomCarrierStalkCommRingCatPresheafCoeContinuousMapCarrierCarrierHomTopCatBaseRingHomHomStalkMap, TopCat.Presheaf.submonoidPresheafOfStalk_obj, AlgebraicGeometry.StructureSheaf.algebraMap_pushforward_stalk, AlgebraicGeometry.sourceAffineLocally_morphismRestrict, RingHom.RespectsIso.arrow_mk_iso_iff, AlgebraicGeometry.Scheme.IdealSheafData.ideal_map_of_isAffineHom, AlgebraicGeometry.LocallyRingedSpace.isLocalHomValStalkMap, AlgebraicGeometry.localRingHom_comp_stalkIso, AlgebraicGeometry.Scheme.localRingHom_comp_stalkIso, AlgebraicGeometry.Scheme.IdealSheafData.map_ideal_basicOpen, AlgebraicGeometry.Scheme.localRingHom_comp_stalkIso_apply, AlgebraicGeometry.Scheme.Hom.isOpen_smoothLocus, AlgebraicGeometry.Scheme.Hom.fromNormalization_app, AlgebraicGeometry.flat_iff, AlgebraicGeometry.Scheme.zeroLocus_map_of_eq, AlgebraicGeometry.HasRingHomProperty.iff_of_isAffine, AlgebraicGeometry.IsAffineOpen.appLE_eq_away_map, AlgebraicGeometry.Spec.map_app, AlgebraicGeometry.IsAffineOpen.ideal_ext_iff, AlgebraicGeometry.Spec.map_base_apply, AlgebraicGeometry.IsAffineOpen.mem_ideal_iff, AlgebraicGeometry.Smooth.iff_forall_exists_isStandardSmooth
hom' 📖	CompOp	3 math math: AlgebraicGeometry.ProjectiveSpectrum.Proj.awayToSection_apply, AlgebraicGeometry.StructureSheaf.toPushforwardStalkAlgHom_apply, ext_iff

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
ext 📖	—	hom'	—	—	—
ext_iff 📖	mathematical	—	hom'	—	ext

CommRingCat.Hom.Simps

Definitions

Name	Category	Theorems
hom 📖	CompOp	—

CommSemiRingCat

Definitions

Name	Category	Theorems
carrier 📖	CompOp	23 math math: forget_obj, hom_inv_apply, CommRingCat.forget₂CommSemiRing_preservesLimitsOfSize, hom_id, ofHom_apply, forgetReflectIsos, coe_of, forget₂SemiRing_preservesLimits, forget_preservesLimits, CommRingCat.forget₂CommSemiRing_preservesLimits, hom_comp, FilteredColimits.forget_preservesFilteredColimits, comp_apply, id_apply, ofHom_hom, forget_map, FilteredColimits.forget₂SemiRing_preservesFilteredColimits, forget₂SemiRing_preservesLimitsOfSize, forget_preservesLimitsOfSize, inv_hom_apply, instFullSemiRingCatForget₂RingHomCarrierCarrier, of_carrier, CategoryTheory.Iso.commSemiRingCatIsoToRingEquiv_toRingHom
commSemiring 📖	CompOp	22 math math: forget_obj, hom_inv_apply, CommRingCat.forget₂CommSemiRing_preservesLimitsOfSize, hom_id, ofHom_apply, forgetReflectIsos, forget₂SemiRing_preservesLimits, forget_preservesLimits, CommRingCat.forget₂CommSemiRing_preservesLimits, hom_comp, FilteredColimits.forget_preservesFilteredColimits, comp_apply, id_apply, ofHom_hom, forget_map, FilteredColimits.forget₂SemiRing_preservesFilteredColimits, forget₂SemiRing_preservesLimitsOfSize, forget_preservesLimitsOfSize, inv_hom_apply, instFullSemiRingCatForget₂RingHomCarrierCarrier, of_carrier, CategoryTheory.Iso.commSemiRingCatIsoToRingEquiv_toRingHom
forget_obj_eq_coe 📖	MathDef	—
fullyFaithfulForget₂ToSemiRingCat 📖	CompOp	—
hasForgetToCommMonCat 📖	CompOp	—
hasForgetToSemiRingCat 📖	CompOp	4 math math: forget₂SemiRing_preservesLimits, FilteredColimits.forget₂SemiRing_preservesFilteredColimits, forget₂SemiRing_preservesLimitsOfSize, instFullSemiRingCatForget₂RingHomCarrierCarrier
instCategory 📖	CompOp	28 math math: forget_obj, hom_inv_apply, CommRingCat.forget₂CommSemiRing_preservesLimitsOfSize, hom_id, RingEquiv.toCommSemiRingCatIso_inv, ofHom_comp, hasLimitsOfSize, ofHom_apply, ofHom_id, hasLimit, forgetReflectIsos, forget₂SemiRing_preservesLimits, forget_preservesLimits, CommRingCat.forget₂CommSemiRing_preservesLimits, hom_comp, FilteredColimits.forget_preservesFilteredColimits, RingEquiv.toCommSemiRingCatIso_hom, comp_apply, id_apply, forget_map, FilteredColimits.forget₂SemiRing_preservesFilteredColimits, forget₂SemiRing_preservesLimitsOfSize, forget_preservesLimitsOfSize, hasLimits, inv_hom_apply, hasLimitsOfShape, instFullSemiRingCatForget₂RingHomCarrierCarrier, CategoryTheory.Iso.commSemiRingCatIsoToRingEquiv_toRingHom
instCoeSortType 📖	CompOp	—
instCommSemiringObjForgetRingHomCarrier 📖	CompOp	—
instConcreteCategoryRingHomCarrier 📖	CompOp	17 math math: forget_obj, hom_inv_apply, CommRingCat.forget₂CommSemiRing_preservesLimitsOfSize, ofHom_apply, forgetReflectIsos, forget₂SemiRing_preservesLimits, forget_preservesLimits, CommRingCat.forget₂CommSemiRing_preservesLimits, FilteredColimits.forget_preservesFilteredColimits, comp_apply, id_apply, forget_map, FilteredColimits.forget₂SemiRing_preservesFilteredColimits, forget₂SemiRing_preservesLimitsOfSize, forget_preservesLimitsOfSize, inv_hom_apply, instFullSemiRingCatForget₂RingHomCarrierCarrier
instInhabited 📖	CompOp	—
ofHom 📖	CompOp	7 math math: RingEquiv.toCommSemiRingCatIso_inv, ofHom_comp, hom_ofHom, ofHom_apply, ofHom_id, RingEquiv.toCommSemiRingCatIso_hom, ofHom_hom

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
coe_of 📖	mathematical	—	carrier of	—	—
comp_apply 📖	mathematical	—	DFunLike.coe carrier CategoryTheory.ConcreteCategory.hom CommSemiRingCat instCategory RingHom Semiring.toNonAssocSemiring CommSemiring.toSemiring commSemiring RingHom.instFunLike instConcreteCategoryRingHomCarrier CategoryTheory.CategoryStruct.comp CategoryTheory.Category.toCategoryStruct	—	—
forgetReflectIsos 📖	mathematical	—	CategoryTheory.Functor.ReflectsIsomorphisms CommSemiRingCat instCategory CategoryTheory.types CategoryTheory.forget RingHom carrier Semiring.toNonAssocSemiring CommSemiring.toSemiring commSemiring RingHom.instFunLike instConcreteCategoryRingHomCarrier	—	Equiv.left_inv Equiv.right_inv MonoidHom.map_mul' RingHom.map_add' CategoryTheory.Iso.isIso_hom
forget_map 📖	mathematical	—	CategoryTheory.Functor.map CommSemiRingCat instCategory CategoryTheory.types CategoryTheory.forget RingHom carrier Semiring.toNonAssocSemiring CommSemiring.toSemiring commSemiring RingHom.instFunLike instConcreteCategoryRingHomCarrier DFunLike.coe CategoryTheory.ConcreteCategory.hom	—	—
forget_obj 📖	mathematical	—	CategoryTheory.Functor.obj CommSemiRingCat instCategory CategoryTheory.types CategoryTheory.forget RingHom carrier Semiring.toNonAssocSemiring CommSemiring.toSemiring commSemiring RingHom.instFunLike instConcreteCategoryRingHomCarrier	—	—
hom_comp 📖	mathematical	—	Hom.hom CategoryTheory.CategoryStruct.comp CommSemiRingCat CategoryTheory.Category.toCategoryStruct instCategory RingHom.comp carrier Semiring.toNonAssocSemiring CommSemiring.toSemiring commSemiring	—	—
hom_ext 📖	—	Hom.hom	—	—	Hom.ext
hom_ext_iff 📖	mathematical	—	Hom.hom	—	hom_ext
hom_id 📖	mathematical	—	Hom.hom CategoryTheory.CategoryStruct.id CommSemiRingCat CategoryTheory.Category.toCategoryStruct instCategory RingHom.id carrier Semiring.toNonAssocSemiring CommSemiring.toSemiring commSemiring	—	—
hom_inv_apply 📖	mathematical	—	DFunLike.coe carrier CategoryTheory.ConcreteCategory.hom CommSemiRingCat instCategory RingHom Semiring.toNonAssocSemiring CommSemiring.toSemiring commSemiring RingHom.instFunLike instConcreteCategoryRingHomCarrier CategoryTheory.Iso.hom CategoryTheory.Iso.inv	—	CategoryTheory.Iso.inv_hom_id_apply
hom_ofHom 📖	mathematical	—	Hom.hom of ofHom	—	—
id_apply 📖	mathematical	—	DFunLike.coe carrier CategoryTheory.ConcreteCategory.hom CommSemiRingCat instCategory RingHom Semiring.toNonAssocSemiring CommSemiring.toSemiring commSemiring RingHom.instFunLike instConcreteCategoryRingHomCarrier CategoryTheory.CategoryStruct.id CategoryTheory.Category.toCategoryStruct	—	—
instFullSemiRingCatForget₂RingHomCarrierCarrier 📖	mathematical	—	CategoryTheory.Functor.Full CommSemiRingCat instCategory SemiRingCat SemiRingCat.instCategory CategoryTheory.forget₂ RingHom carrier Semiring.toNonAssocSemiring CommSemiring.toSemiring commSemiring RingHom.instFunLike instConcreteCategoryRingHomCarrier SemiRingCat.carrier SemiRingCat.semiring SemiRingCat.instConcreteCategoryRingHomCarrier hasForgetToSemiRingCat	—	CategoryTheory.Functor.FullyFaithful.full
inv_hom_apply 📖	mathematical	—	DFunLike.coe carrier CategoryTheory.ConcreteCategory.hom CommSemiRingCat instCategory RingHom Semiring.toNonAssocSemiring CommSemiring.toSemiring commSemiring RingHom.instFunLike instConcreteCategoryRingHomCarrier CategoryTheory.Iso.inv CategoryTheory.Iso.hom	—	CategoryTheory.Iso.hom_inv_id_apply
ofHom_apply 📖	mathematical	—	DFunLike.coe of carrier CategoryTheory.ConcreteCategory.hom CommSemiRingCat instCategory RingHom Semiring.toNonAssocSemiring CommSemiring.toSemiring commSemiring RingHom.instFunLike instConcreteCategoryRingHomCarrier ofHom	—	—
ofHom_comp 📖	mathematical	—	ofHom RingHom.comp Semiring.toNonAssocSemiring CommSemiring.toSemiring CategoryTheory.CategoryStruct.comp CommSemiRingCat CategoryTheory.Category.toCategoryStruct instCategory of	—	—
ofHom_hom 📖	mathematical	—	ofHom carrier commSemiring Hom.hom	—	—
ofHom_id 📖	mathematical	—	ofHom RingHom.id Semiring.toNonAssocSemiring CommSemiring.toSemiring CategoryTheory.CategoryStruct.id CommSemiRingCat CategoryTheory.Category.toCategoryStruct instCategory of	—	—
of_carrier 📖	mathematical	—	of carrier commSemiring	—	—

CommSemiRingCat.Hom

Definitions

Name	Category	Theorems
hom 📖	CompOp	6 math math: CommSemiRingCat.hom_id, CommSemiRingCat.hom_ofHom, CommSemiRingCat.hom_ext_iff, CommSemiRingCat.hom_comp, CommSemiRingCat.ofHom_hom, CategoryTheory.Iso.commSemiRingCatIsoToRingEquiv_toRingHom
hom' 📖	CompOp	1 math math: ext_iff

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
ext 📖	—	hom'	—	—	—
ext_iff 📖	mathematical	—	hom'	—	ext

CommSemiRingCat.Hom.Simps

Definitions

Name	Category	Theorems
hom 📖	CompOp	—

RingCat

Definitions

Name	Category	Theorems
carrier 📖	CompOp	161 math math: PresheafOfModules.Monoidal.tensorObj_obj, CommRingCat.forget₂Ring_preservesLimitsOfSize, AlgebraicGeometry.StructureSheaf.instIsScalarTowerCarrierStalkCommRingCatStructurePresheafInCommRingCatCarrierAbPresheafOpensCarrierTopModuleStructurePresheaf, CommRingCat.instFullRingCatForget₂RingHomCarrierCarrier, FilteredColimits.instPreservesFilteredColimitsAddCommGrpCatForget₂RingHomCarrierAddMonoidHomCarrier, PresheafOfModules.add_app, PresheafOfModules.Sheafify.add_smul, PresheafOfModules.restrictScalars_map_app, PresheafOfModules.epi_iff_surjective, PresheafOfModules.evaluation_preservesColimitsOfShape, PresheafOfModules.comp_app, Colimits.cocone_naturality_components, PresheafOfModules.pushforward_map_app_apply, PresheafOfModules.sections_property, forget₂AddCommGroup_preservesLimits, PresheafOfModules.toSheafify_app_apply', PresheafOfModules.instPreservesLimitsOfShapeModuleCatCarrierObjOppositeRingCatEvaluation, PresheafOfModules.toFreeYonedaCoproduct_fromFreeYonedaCoproduct, PresheafOfModules.Derivation.d_map, PresheafOfModules.free_map_app, forget₂_map, PresheafOfModules.congr_map_apply, hom_comp, PresheafOfModules.freeYonedaEquiv_symm_app, PresheafOfModules.restrictScalarsObj_map, PresheafOfModules.forgetToPresheafModuleCatObj_map, inv_hom_apply, SheafOfModules.evaluationPreservesLimitsOfShape, PresheafOfModules.pushforward₀_obj_obj_carrier, PresheafOfModules.id_app, PresheafOfModules.restrictScalarsObj_obj, PresheafOfModules.Derivation.postcomp_d_apply, forget_map_apply, PresheafOfModules.Derivation.d_one, PresheafOfModules.sectionsMap_coe, PresheafOfModules.map_comp_apply, PresheafOfModules.pushforward_map_app_apply', PresheafOfModules.Derivation.d_mul, SheafOfModules.evaluationPreservesLimitsOfSize, PresheafOfModules.unit_map_one, PresheafOfModules.zsmul_app, PresheafOfModules.toPresheaf_obj_coe, AlgCat.forget₂Ring_preservesLimitsOfSize, PresheafOfModules.isoMk_hom_app, PresheafOfModules.map_smul, PresheafOfModules.Monoidal.tensorObj_map_tmul, PresheafOfModules.instPreservesLimitsOfSizeModuleCatCarrierObjOppositeRingCatEvaluation, of_carrier, SheafOfModules.pushforwardComp_inv_app_val_app, forget₂SemiRing_preservesLimitsOfSize, PresheafOfModules.pushforward₀_obj_obj_isAddCommGroup, PresheafOfModules.mono_iff_surjective, PresheafOfModules.freeObj_map, PresheafOfModules.instEpiModuleCatCarrierObjOppositeRingCatApp, PresheafOfModules.presheaf_map_apply_coe, PresheafOfModules.Derivation.congr_d, PresheafOfModules.forgetToPresheafModuleCatObjMap_apply, SheafOfModules.pushforwardCongr_hom_app_val_app, FilteredColimits.forget_preservesFilteredColimits, CategoryTheory.Iso.ringCatIsoToRingEquiv_toRingHom, PresheafOfModules.toFreeYonedaCoproduct_fromFreeYonedaCoproduct_assoc, PresheafOfModules.surjective_of_epi, PresheafOfModules.injective_of_mono, PresheafOfModules.isoMk_inv_app, PresheafOfModules.pushforward_obj_map_apply, forget_preservesLimitsOfSize, PresheafOfModules.forgetToPresheafModuleCatObjObj_coe, ofHom_apply, PresheafOfModules.unitHomEquiv_apply_coe, PresheafOfModules.freeObj_obj, SheafOfModules.pushforwardCongr_inv_app_val_app, CommRingCat.forgetToRingCat_obj, moduleCatRestrictScalarsPseudofunctor_mapId, PresheafOfModules.freeAdjunctionUnit_app, SheafOfModules.pushforwardPushforwardAdj_unit_app_val_app, PresheafOfModules.presheaf_obj_coe, PresheafOfModules.toPresheaf_map_app_apply, PresheafOfModules.Derivation'.d_app, PresheafOfModules.neg_app, PresheafOfModules.map_comp, moduleCatRestrictScalarsPseudofunctor_map, PresheafOfModules.evaluation_preservesColimitsOfSize, SheafOfModules.pushforwardComp_hom_app_val_app, PresheafOfModules.evaluation_preservesFiniteLimits, SheafOfModules.Presentation.map_relations_I, PresheafOfModules.sub_app, PresheafOfModules.DifferentialsConstruction.relativeDifferentials'_map_d, PresheafOfModules.Elements.fromFreeYoneda_app_apply, PresheafOfModules.pushforward_obj_obj, SheafOfModules.Presentation.mapRelations_mapGenerators, SheafOfModules.relationsOfIsCokernelFree_I, CommRingCat.forget₂Ring_preservesLimits, forgetReflectIsos, PresheafOfModules.instAdditiveModuleCatCarrierObjOppositeRingCatEvaluation, PresheafOfModules.Hom.naturality_assoc, CommRingCat.FilteredColimits.forget₂Ring_preservesFilteredColimits, PresheafOfModules.restriction_app, PresheafOfModules.ι_fromFreeYonedaCoproduct_apply, PresheafOfModules.DifferentialsConstruction.relativeDifferentials'_obj, PresheafOfModules.Derivation.d_app, PresheafOfModules.forgetToPresheafModuleCat_obj, SheafOfModules.unitToPushforwardObjUnit_val_app_apply, AlgebraicGeometry.instIsIsoModulesSpecOfCarrierFromTildeΓUnitOpensCarrierCarrierCommRingCatRingCatSheaf, PresheafOfModules.forgetToPresheafModuleCatObj_obj, PresheafOfModules.Monoidal.tensorHom_app, instFullSemiRingCatForget₂RingHomCarrierCarrier, PresheafOfModules.DifferentialsConstruction.relativeDifferentials'_map, moduleCatRestrictScalarsPseudofunctor_obj, PresheafOfModules.instMonoModuleCatCarrierObjOppositeRingCatApp, PresheafOfModules.Derivation'.app_apply, AlgebraicGeometry.StructureSheaf.instIsLocalizedModuleCarrierStalkAbPresheafOpensCarrierTopModuleStructurePresheafPrimeComplAsIdealToStalkₗ, SheafOfModules.Presentation.mapRelations_mapGenerators_assoc, hom_id, moduleCatRestrictScalarsPseudofunctor_mapComp, forget₂SemiRing_preservesLimits, comp_apply, SheafOfModules.pushforwardNatTrans_app_val_app, PresheafOfModules.Sheafify.one_smul, forget₂AddCommGroup_preservesLimitsOfSize, PresheafOfModules.map_id, ofHom_hom, SheafOfModules.pushforwardPushforwardEquivalence_unit_app_val_app, SheafOfModules.relationsOfIsCokernelFree_s, coe_of, PresheafOfModules.pushforward_obj_map_apply', PresheafOfModules.Hom.naturality, AlgCat.forget₂Ring_preservesLimits, forget_preservesLimits, SheafOfModules.forgetToSheafModuleCat_map_val, SheafOfModules.pushforwardPushforwardAdj_counit_app_val_app, PresheafOfModules.freeYonedaEquiv_comp, PresheafOfModules.Sheafify.zero_smul, AlgebraicGeometry.instIsQuasicoherentOpensCarrierCarrierCommRingCatSpecTilde, SheafOfModules.Presentation.IsFinite.finite_relations, SheafOfModules.pushforwardNatTrans_app_val_app_apply, PresheafOfModules.Finite.evaluation_preservesFiniteColimits, hom_inv_apply, CommRingCat.forgetToRingCat_map_hom, PresheafOfModules.germ_ringCat_smul, PresheafOfModules.naturality_apply, PresheafOfModules.fromFreeYonedaCoproduct_app_mk, PresheafOfModules.Sheafify.map_smul, PresheafOfModules.germ_smul, PresheafOfModules.toSheafify_app_apply, PresheafOfModules.pushforward₀_obj_obj_isModule, id_apply, SheafOfModules.forgetToSheafModuleCat_obj_val, PresheafOfModules.evaluation_obj, forget_obj, FilteredColimits.forget₂SemiRing_preservesFilteredColimits, SheafOfModules.pushforwardPushforwardEquivalence_counit_app_val_app, PresheafOfModules.evaluation_map, PresheafOfModules.freeObjDesc_app, PresheafOfModules.Sheafify.mul_smul, AlgebraicGeometry.instIsIsoModulesSpecOfCarrierFromTildeΓFreeOpensCarrierCarrierCommRingCat, PresheafOfModules.zero_app, PresheafOfModules.map_comp_assoc, PresheafOfModules.forgetToPresheafModuleCat_map, SheafOfModules.Finite.evaluationPreservesFiniteLimits, forget_map, SheafOfModules.unitHomEquiv_apply_coe
forget_obj_eq_coe 📖	MathDef	—
fullyFaithfulForget₂ToSemiRingCat 📖	CompOp	—
hasForgetToAddCommGrp 📖	CompOp	6 math math: FilteredColimits.instPreservesFilteredColimitsAddCommGrpCatForget₂RingHomCarrierAddMonoidHomCarrier, forget₂AddCommGroup_preservesLimits, AlgebraicGeometry.instIsIsoModulesSpecOfCarrierFromTildeΓUnitOpensCarrierCarrierCommRingCatRingCatSheaf, forget₂AddCommGroup_preservesLimitsOfSize, AlgebraicGeometry.instIsQuasicoherentOpensCarrierCarrierCommRingCatSpecTilde, AlgebraicGeometry.instIsIsoModulesSpecOfCarrierFromTildeΓFreeOpensCarrierCarrierCommRingCat
hasForgetToSemiRingCat 📖	CompOp	5 math math: forget₂_map, forget₂SemiRing_preservesLimitsOfSize, instFullSemiRingCatForget₂RingHomCarrierCarrier, forget₂SemiRing_preservesLimits, FilteredColimits.forget₂SemiRing_preservesFilteredColimits
instCategory 📖	CompOp	225 math math: PresheafOfModules.Monoidal.tensorObj_obj, PresheafOfModules.instIsRightAdjointPushforwardCompFunctorOppositeRingCatWhiskerLeftOp, SheafOfModules.pushforward_assoc, CommRingCat.forget₂Ring_preservesLimitsOfSize, AlgebraicGeometry.StructureSheaf.instIsScalarTowerCarrierStalkCommRingCatStructurePresheafInCommRingCatCarrierAbPresheafOpensCarrierTopModuleStructurePresheaf, RingEquiv.toRingCatIso_inv, CommRingCat.instFullRingCatForget₂RingHomCarrierCarrier, AlgebraicGeometry.Scheme.Modules.instFaithfulPresheafOfModulesToPresheafOfModules, FilteredColimits.instPreservesFilteredColimitsAddCommGrpCatForget₂RingHomCarrierAddMonoidHomCarrier, PresheafOfModules.pullback_id_comp, PresheafOfModules.instIsLocallySurjectiveToSheafify, PresheafOfModules.add_app, SheafOfModules.pushforwardNatIso_inv, PresheafOfModules.Sheafify.add_smul, PresheafOfModules.restrictScalars_map_app, SheafOfModules.instFullPresheafOfModulesValRingCatForget, PresheafOfModules.epi_iff_surjective, PresheafOfModules.pullback_comp_id, PresheafOfModules.evaluation_preservesColimitsOfShape, PresheafOfModules.comp_app, Colimits.cocone_naturality_components, PresheafOfModules.pushforward_map_app_apply, PresheafOfModules.sections_property, forget₂AddCommGroup_preservesLimits, PresheafOfModules.toSheafify_app_apply', PresheafOfModules.instPreservesLimitsOfShapeModuleCatCarrierObjOppositeRingCatEvaluation, PresheafOfModules.toFreeYonedaCoproduct_fromFreeYonedaCoproduct, PresheafOfModules.Derivation.d_map, PresheafOfModules.free_map_app, forget₂_map, SheafOfModules.instIsRightAdjointPushforwardCompSheafRingCatMapSheafPushforwardContinuous, PresheafOfModules.congr_map_apply, hom_comp, PresheafOfModules.freeYonedaEquiv_symm_app, PresheafOfModules.restrictScalarsObj_map, PresheafOfModules.forgetToPresheafModuleCatObj_map, inv_hom_apply, SheafOfModules.evaluationPreservesLimitsOfShape, SheafOfModules.toSheaf_obj_val, PresheafOfModules.pushforward₀_obj_obj_carrier, PresheafOfModules.id_app, PresheafOfModules.toPresheaf_map_toSheafify, PresheafOfModules.restrictScalarsObj_obj, PresheafOfModules.Derivation.postcomp_d_apply, forget_map_apply, PresheafOfModules.Derivation.d_one, PresheafOfModules.sectionsMap_coe, PresheafOfModules.map_comp_apply, PresheafOfModules.pushforward_map_app_apply', PresheafOfModules.Derivation.d_mul, SheafOfModules.evaluationPreservesLimitsOfSize, PresheafOfModules.unit_map_one, PresheafOfModules.zsmul_app, PresheafOfModules.toPresheaf_obj_coe, AlgCat.forget₂Ring_preservesLimitsOfSize, PresheafOfModules.isoMk_hom_app, SheafOfModules.isSheaf, PresheafOfModules.map_smul, PresheafOfModules.Monoidal.tensorObj_map_tmul, hasLimitsOfShape, PresheafOfModules.instPreservesLimitsOfSizeModuleCatCarrierObjOppositeRingCatEvaluation, SheafOfModules.pushforwardComp_inv_app_val_app, forget₂SemiRing_preservesLimitsOfSize, PresheafOfModules.pushforward₀_obj_obj_isAddCommGroup, Colimits.cocone_naturality, SheafOfModules.pushforwardNatTrans_id, SheafOfModules.restrictScalars_obj_val, PresheafOfModules.mono_iff_surjective, PresheafOfModules.freeObj_map, SheafOfModules.instReflectsIsomorphismsPresheafOfModulesValRingCatForget, PresheafOfModules.instEpiModuleCatCarrierObjOppositeRingCatApp, PresheafOfModules.presheaf_map_apply_coe, SheafOfModules.instFaithfulPresheafOfModulesValRingCatForget, PresheafOfModules.Derivation.congr_d, RingEquiv.toRingCatIso_hom, PresheafOfModules.forgetToPresheafModuleCatObjMap_apply, SheafOfModules.pushforwardCongr_hom_app_val_app, SheafOfModules.toSheaf_map_val, PresheafOfModules.pullback_assoc, FilteredColimits.forget_preservesFilteredColimits, CategoryTheory.Iso.ringCatIsoToRingEquiv_toRingHom, PresheafOfModules.toFreeYonedaCoproduct_fromFreeYonedaCoproduct_assoc, SheafOfModules.pullback_comp_id, PresheafOfModules.surjective_of_epi, PresheafOfModules.toSheaf_map_sheafificationHomEquiv_symm, PresheafOfModules.injective_of_mono, PresheafOfModules.isoMk_inv_app, SheafOfModules.id_val, PresheafOfModules.pushforward_obj_map_apply, forget_preservesLimitsOfSize, PresheafOfModules.forgetToPresheafModuleCatObjObj_coe, hasLimits, ofHom_apply, PresheafOfModules.unitHomEquiv_apply_coe, PresheafOfModules.freeObj_obj, SheafOfModules.pushforwardCongr_inv_app_val_app, CommRingCat.forgetToRingCat_obj, SheafOfModules.conjugateEquiv_pullbackComp_inv, moduleCatRestrictScalarsPseudofunctor_mapId, PresheafOfModules.freeAdjunctionUnit_app, PresheafOfModules.presheaf_obj_coe, SheafOfModules.fullyFaithfulForget_preimage_val, SheafOfModules.pullback_id_comp, SheafOfModules.restrictScalars_map_val, Colimits.hasColimits_ringCat, PresheafOfModules.toPresheaf_map_app_apply, hasLimitsOfSize, PresheafOfModules.Derivation'.d_app, PresheafOfModules.neg_app, PresheafOfModules.map_comp, moduleCatRestrictScalarsPseudofunctor_map, SheafOfModules.pushforward_map_val, PresheafOfModules.evaluation_preservesColimitsOfSize, SheafOfModules.pushforwardComp_hom_app_val_app, SheafOfModules.unit_val, PresheafOfModules.evaluation_preservesFiniteLimits, SheafOfModules.Presentation.map_relations_I, SheafOfModules.instIsLeftAdjointOverOverRingCatPushforwardIdSheafOver, PresheafOfModules.sub_app, PresheafOfModules.DifferentialsConstruction.relativeDifferentials'_map_d, PresheafOfModules.Elements.fromFreeYoneda_app_apply, PresheafOfModules.pushforward_obj_obj, SheafOfModules.Presentation.mapRelations_mapGenerators, SheafOfModules.relationsOfIsCokernelFree_I, CommRingCat.forget₂Ring_preservesLimits, forgetReflectIsos, SheafOfModules.pushforwardCongr_symm, PresheafOfModules.instAdditiveModuleCatCarrierObjOppositeRingCatEvaluation, PresheafOfModules.Hom.naturality_assoc, ofHom_id, SheafOfModules.add_val, PresheafOfModules.sheafifyMap_val, CommRingCat.FilteredColimits.forget₂Ring_preservesFilteredColimits, PresheafOfModules.restriction_app, PresheafOfModules.ι_fromFreeYonedaCoproduct_apply, PresheafOfModules.DifferentialsConstruction.relativeDifferentials'_obj, PresheafOfModules.Derivation.d_app, PresheafOfModules.forgetToPresheafModuleCat_obj, SheafOfModules.unitToPushforwardObjUnit_val_app_apply, AlgebraicGeometry.instIsIsoModulesSpecOfCarrierFromTildeΓUnitOpensCarrierCarrierCommRingCatRingCatSheaf, PresheafOfModules.forgetToPresheafModuleCatObj_obj, SheafOfModules.pullback_assoc, SheafOfModules.forget_map, PresheafOfModules.Monoidal.tensorHom_app, instFullSemiRingCatForget₂RingHomCarrierCarrier, PresheafOfModules.DifferentialsConstruction.relativeDifferentials'_map, SheafOfModules.forgetPreservesLimitsOfSize, moduleCatRestrictScalarsPseudofunctor_obj, PresheafOfModules.instMonoModuleCatCarrierObjOppositeRingCatApp, PresheafOfModules.Derivation'.app_apply, AlgebraicGeometry.StructureSheaf.instIsLocalizedModuleCarrierStalkAbPresheafOpensCarrierTopModuleStructurePresheafPrimeComplAsIdealToStalkₗ, SheafOfModules.Finite.forgetPreservesFiniteLimits, SheafOfModules.Presentation.mapRelations_mapGenerators_assoc, SheafOfModules.forget_obj, hom_id, moduleCatRestrictScalarsPseudofunctor_mapComp, SheafOfModules.comp_val_assoc, forget₂SemiRing_preservesLimits, ofHom_comp, comp_apply, SheafOfModules.pushforwardNatTrans_app_val_app, PresheafOfModules.Sheafify.one_smul, forget₂AddCommGroup_preservesLimitsOfSize, PresheafOfModules.pushforward_assoc, PresheafOfModules.map_id, AlgebraicGeometry.Scheme.Modules.instFullPresheafOfModulesToPresheafOfModules, SheafOfModules.pushforward_comp_id, SheafOfModules.relationsOfIsCokernelFree_s, PresheafOfModules.pushforward_obj_map_apply', PresheafOfModules.Hom.naturality, AlgCat.forget₂Ring_preservesLimits, forget_preservesLimits, SheafOfModules.forgetToSheafModuleCat_map_val, SheafOfModules.conjugateEquiv_pullbackId_hom, hasLimit, PresheafOfModules.freeYonedaEquiv_comp, PresheafOfModules.Sheafify.zero_smul, PresheafOfModules.sheafificationAdjunction_homEquiv_apply, AlgebraicGeometry.instIsQuasicoherentOpensCarrierCarrierCommRingCatSpecTilde, SheafOfModules.forgetPreservesLimitsOfShape, SheafOfModules.Presentation.IsFinite.finite_relations, SheafOfModules.pushforwardNatTrans_app_val_app_apply, PresheafOfModules.Finite.evaluation_preservesFiniteColimits, SheafOfModules.comp_val, PresheafOfModules.toPresheaf_map_sheafificationAdjunction_unit_app, SheafOfModules.pushforward_id_comp, hom_inv_apply, PresheafOfModules.instIsLocallyInjectiveToSheafify, CommRingCat.forgetToRingCat_map_hom, PresheafOfModules.germ_ringCat_smul, AlgebraicGeometry.Scheme.Modules.instIsRightAdjointPresheafOfModulesToPresheafOfModules, PresheafOfModules.naturality_apply, PresheafOfModules.toPresheaf_map_sheafificationHomEquiv_def, PresheafOfModules.instFullRestrictScalarsIdFunctorOppositeRingCat, SheafOfModules.pushforwardSections_coe, PresheafOfModules.fromFreeYonedaCoproduct_app_mk, PresheafOfModules.Sheafify.map_smul, PresheafOfModules.germ_smul, PresheafOfModules.toSheafify_app_apply, SheafOfModules.instIsRightAdjointPushforwardIdSheafRingCat, PresheafOfModules.pushforward₀_obj_obj_isModule, id_apply, SheafOfModules.forgetToSheafModuleCat_obj_val, PresheafOfModules.evaluation_obj, SheafOfModules.pushforwardNatIso_hom, forget_obj, PresheafOfModules.pushforward_id_comp, FilteredColimits.forget₂SemiRing_preservesFilteredColimits, PresheafOfModules.evaluation_map, PresheafOfModules.freeObjDesc_app, SheafOfModules.pushforward_obj_val, PresheafOfModules.Sheafify.mul_smul, AlgebraicGeometry.instIsIsoModulesSpecOfCarrierFromTildeΓFreeOpensCarrierCarrierCommRingCat, PresheafOfModules.zero_app, PresheafOfModules.map_comp_assoc, PresheafOfModules.forgetToPresheafModuleCat_map, PresheafOfModules.toPresheaf_map_sheafificationHomEquiv, PresheafOfModules.instIsRightAdjointPushforwardIdFunctorOppositeRingCat, SheafOfModules.Finite.evaluationPreservesFiniteLimits, SheafOfModules.instAdditivePresheafOfModulesValRingCatForget, PresheafOfModules.pushforward_comp_id, SheafOfModules.pushforwardNatTrans_comp, forget_map, PresheafOfModules.pushforward₀_obj_map, SheafOfModules.unitHomEquiv_apply_coe
instCoeSortType 📖	CompOp	—
instConcreteCategoryRingHomCarrier 📖	CompOp	53 math math: PresheafOfModules.Monoidal.tensorObj_obj, CommRingCat.forget₂Ring_preservesLimitsOfSize, AlgebraicGeometry.StructureSheaf.instIsScalarTowerCarrierStalkCommRingCatStructurePresheafInCommRingCatCarrierAbPresheafOpensCarrierTopModuleStructurePresheaf, CommRingCat.instFullRingCatForget₂RingHomCarrierCarrier, FilteredColimits.instPreservesFilteredColimitsAddCommGrpCatForget₂RingHomCarrierAddMonoidHomCarrier, Colimits.cocone_naturality_components, forget₂AddCommGroup_preservesLimits, PresheafOfModules.Derivation.d_map, forget₂_map, inv_hom_apply, PresheafOfModules.Derivation.postcomp_d_apply, forget_map_apply, PresheafOfModules.Derivation.d_one, PresheafOfModules.Derivation.d_mul, AlgCat.forget₂Ring_preservesLimitsOfSize, PresheafOfModules.map_smul, PresheafOfModules.Monoidal.tensorObj_map_tmul, forget₂SemiRing_preservesLimitsOfSize, PresheafOfModules.Derivation.congr_d, FilteredColimits.forget_preservesFilteredColimits, forget_preservesLimitsOfSize, ofHom_apply, CommRingCat.forgetToRingCat_obj, PresheafOfModules.Derivation'.d_app, PresheafOfModules.DifferentialsConstruction.relativeDifferentials'_map_d, CommRingCat.forget₂Ring_preservesLimits, forgetReflectIsos, CommRingCat.FilteredColimits.forget₂Ring_preservesFilteredColimits, PresheafOfModules.DifferentialsConstruction.relativeDifferentials'_obj, PresheafOfModules.Derivation.d_app, SheafOfModules.unitToPushforwardObjUnit_val_app_apply, AlgebraicGeometry.instIsIsoModulesSpecOfCarrierFromTildeΓUnitOpensCarrierCarrierCommRingCatRingCatSheaf, PresheafOfModules.Monoidal.tensorHom_app, instFullSemiRingCatForget₂RingHomCarrierCarrier, PresheafOfModules.DifferentialsConstruction.relativeDifferentials'_map, PresheafOfModules.Derivation'.app_apply, AlgebraicGeometry.StructureSheaf.instIsLocalizedModuleCarrierStalkAbPresheafOpensCarrierTopModuleStructurePresheafPrimeComplAsIdealToStalkₗ, forget₂SemiRing_preservesLimits, comp_apply, forget₂AddCommGroup_preservesLimitsOfSize, AlgCat.forget₂Ring_preservesLimits, forget_preservesLimits, AlgebraicGeometry.instIsQuasicoherentOpensCarrierCarrierCommRingCatSpecTilde, hom_inv_apply, CommRingCat.forgetToRingCat_map_hom, PresheafOfModules.germ_ringCat_smul, PresheafOfModules.Sheafify.map_smul, PresheafOfModules.germ_smul, id_apply, forget_obj, FilteredColimits.forget₂SemiRing_preservesFilteredColimits, AlgebraicGeometry.instIsIsoModulesSpecOfCarrierFromTildeΓFreeOpensCarrierCarrierCommRingCat, forget_map
instInhabited 📖	CompOp	—
instRingObjForgetRingHomCarrier 📖	CompOp	—
ofHom 📖	CompOp	7 math math: RingEquiv.toRingCatIso_inv, hom_ofHom, RingEquiv.toRingCatIso_hom, ofHom_apply, ofHom_id, ofHom_comp, ofHom_hom

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
coe_of 📖	mathematical	—	carrier of	—	—
comp_apply 📖	mathematical	—	DFunLike.coe carrier CategoryTheory.ConcreteCategory.hom RingCat instCategory RingHom Semiring.toNonAssocSemiring Ring.toSemiring ring RingHom.instFunLike instConcreteCategoryRingHomCarrier CategoryTheory.CategoryStruct.comp CategoryTheory.Category.toCategoryStruct	—	—
forgetReflectIsos 📖	mathematical	—	CategoryTheory.Functor.ReflectsIsomorphisms RingCat instCategory CategoryTheory.types CategoryTheory.forget RingHom carrier Semiring.toNonAssocSemiring Ring.toSemiring ring RingHom.instFunLike instConcreteCategoryRingHomCarrier	—	Equiv.left_inv Equiv.right_inv MonoidHom.map_mul' RingHom.map_add' CategoryTheory.Iso.isIso_hom
forget_map 📖	mathematical	—	CategoryTheory.Functor.map RingCat instCategory CategoryTheory.types CategoryTheory.forget RingHom carrier Semiring.toNonAssocSemiring Ring.toSemiring ring RingHom.instFunLike instConcreteCategoryRingHomCarrier DFunLike.coe CategoryTheory.ConcreteCategory.hom	—	—
forget_map_apply 📖	mathematical	—	CategoryTheory.Functor.map RingCat instCategory CategoryTheory.types CategoryTheory.forget RingHom carrier Semiring.toNonAssocSemiring Ring.toSemiring ring RingHom.instFunLike instConcreteCategoryRingHomCarrier DFunLike.coe CategoryTheory.ConcreteCategory.hom	—	—
forget_obj 📖	mathematical	—	CategoryTheory.Functor.obj RingCat instCategory CategoryTheory.types CategoryTheory.forget RingHom carrier Semiring.toNonAssocSemiring Ring.toSemiring ring RingHom.instFunLike instConcreteCategoryRingHomCarrier	—	—
forget₂_map 📖	mathematical	—	DFunLike.coe CategoryTheory.Functor.obj RingCat instCategory SemiRingCat SemiRingCat.instCategory CategoryTheory.forget₂ RingHom carrier Semiring.toNonAssocSemiring Ring.toSemiring ring RingHom.instFunLike instConcreteCategoryRingHomCarrier SemiRingCat.carrier SemiRingCat.semiring SemiRingCat.instConcreteCategoryRingHomCarrier hasForgetToSemiRingCat CategoryTheory.ConcreteCategory.hom CategoryTheory.Functor.map	—	—
hom_comp 📖	mathematical	—	Hom.hom CategoryTheory.CategoryStruct.comp RingCat CategoryTheory.Category.toCategoryStruct instCategory RingHom.comp carrier Semiring.toNonAssocSemiring Ring.toSemiring ring	—	—
hom_ext 📖	—	Hom.hom	—	—	Hom.ext
hom_ext_iff 📖	mathematical	—	Hom.hom	—	hom_ext
hom_id 📖	mathematical	—	Hom.hom CategoryTheory.CategoryStruct.id RingCat CategoryTheory.Category.toCategoryStruct instCategory RingHom.id carrier Semiring.toNonAssocSemiring Ring.toSemiring ring	—	—
hom_inv_apply 📖	mathematical	—	DFunLike.coe carrier CategoryTheory.ConcreteCategory.hom RingCat instCategory RingHom Semiring.toNonAssocSemiring Ring.toSemiring ring RingHom.instFunLike instConcreteCategoryRingHomCarrier CategoryTheory.Iso.hom CategoryTheory.Iso.inv	—	CategoryTheory.Iso.inv_hom_id_apply
hom_ofHom 📖	mathematical	—	Hom.hom of ofHom	—	—
id_apply 📖	mathematical	—	DFunLike.coe carrier CategoryTheory.ConcreteCategory.hom RingCat instCategory RingHom Semiring.toNonAssocSemiring Ring.toSemiring ring RingHom.instFunLike instConcreteCategoryRingHomCarrier CategoryTheory.CategoryStruct.id CategoryTheory.Category.toCategoryStruct	—	—
instFullSemiRingCatForget₂RingHomCarrierCarrier 📖	mathematical	—	CategoryTheory.Functor.Full RingCat instCategory SemiRingCat SemiRingCat.instCategory CategoryTheory.forget₂ RingHom carrier Semiring.toNonAssocSemiring Ring.toSemiring ring RingHom.instFunLike instConcreteCategoryRingHomCarrier SemiRingCat.carrier SemiRingCat.semiring SemiRingCat.instConcreteCategoryRingHomCarrier hasForgetToSemiRingCat	—	CategoryTheory.Functor.FullyFaithful.full
inv_hom_apply 📖	mathematical	—	DFunLike.coe carrier CategoryTheory.ConcreteCategory.hom RingCat instCategory RingHom Semiring.toNonAssocSemiring Ring.toSemiring ring RingHom.instFunLike instConcreteCategoryRingHomCarrier CategoryTheory.Iso.inv CategoryTheory.Iso.hom	—	CategoryTheory.Iso.hom_inv_id_apply
ofHom_apply 📖	mathematical	—	DFunLike.coe of carrier CategoryTheory.ConcreteCategory.hom RingCat instCategory RingHom Semiring.toNonAssocSemiring Ring.toSemiring ring RingHom.instFunLike instConcreteCategoryRingHomCarrier ofHom	—	—
ofHom_comp 📖	mathematical	—	ofHom RingHom.comp Semiring.toNonAssocSemiring Ring.toSemiring CategoryTheory.CategoryStruct.comp RingCat CategoryTheory.Category.toCategoryStruct instCategory of	—	—
ofHom_hom 📖	mathematical	—	ofHom carrier ring Hom.hom	—	—
ofHom_id 📖	mathematical	—	ofHom RingHom.id Semiring.toNonAssocSemiring Ring.toSemiring CategoryTheory.CategoryStruct.id RingCat CategoryTheory.Category.toCategoryStruct instCategory of	—	—
of_carrier 📖	mathematical	—	of carrier ring	—	—

RingCat.Hom

Definitions

Name	Category	Theorems
hom 📖	CompOp	45 math math: PresheafOfModules.restrictScalars_map_app, PresheafOfModules.limitPresheafOfModules_map, PresheafOfModules.sections_property, PresheafOfModules.toSheafify_app_apply', PresheafOfModules.Derivation.d_map, PresheafOfModules.congr_map_apply, RingCat.hom_comp, PresheafOfModules.restrictScalarsObj_map, PresheafOfModules.restrictScalarsObj_obj, PresheafOfModules.map_comp_apply, PresheafOfModules.pushforward_map_app_apply', PresheafOfModules.unit_map_one, PresheafOfModules.map_smul, PresheafOfModules.Monoidal.tensorObj_map_tmul, RingCat.hom_ofHom, PresheafOfModules.freeObj_map, PresheafOfModules.presheaf_map_apply_coe, PresheafOfModules.forgetToPresheafModuleCatObjMap_apply, CategoryTheory.Iso.ringCatIsoToRingEquiv_toRingHom, PresheafOfModules.pushforward_obj_map_apply, SheafOfModules.pushforwardPushforwardAdj_unit_app_val_app, PresheafOfModules.map_comp, RingCat.moduleCatRestrictScalarsPseudofunctor_map, PresheafOfModules.DifferentialsConstruction.relativeDifferentials'_map_d, PresheafOfModules.pushforward_obj_obj, PresheafOfModules.Hom.naturality_assoc, PresheafOfModules.restriction_app, RingCat.hom_ext_iff, RingCat.hom_id, RingCat.moduleCatRestrictScalarsPseudofunctor_mapComp, PresheafOfModules.instHasLimitModuleCatCarrierObjOppositeRingCatCompEvaluationRestrictScalarsHomMap, SheafOfModules.pushforwardNatTrans_app_val_app, PresheafOfModules.map_id, RingCat.ofHom_hom, SheafOfModules.pushforwardPushforwardEquivalence_unit_app_val_app, PresheafOfModules.pushforward_obj_map_apply', PresheafOfModules.Hom.naturality, SheafOfModules.pushforwardPushforwardAdj_counit_app_val_app, CategoryTheory.Presieve.FamilyOfElements.isCompatible_map_smul_aux, SheafOfModules.pushforwardNatTrans_app_val_app_apply, PresheafOfModules.ofPresheaf_map, CommRingCat.forgetToRingCat_map_hom, PresheafOfModules.naturality_apply, SheafOfModules.pushforwardPushforwardEquivalence_counit_app_val_app, PresheafOfModules.map_comp_assoc
hom' 📖	CompOp	1 math math: ext_iff

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
ext 📖	—	hom'	—	—	—
ext_iff 📖	mathematical	—	hom'	—	ext

RingCat.Hom.Simps

Definitions

Name	Category	Theorems
hom 📖	CompOp	—

RingEquiv

Definitions

Name	Category	Theorems
toCommRingCatIso 📖	CompOp	5 math math: AlgebraicGeometry.ProjectiveSpectrum.Proj.awayToΓ_ΓToStalk, toCommRingCatIso_inv, toCommRingCatIso_hom, AlgebraicGeometry.ProjectiveSpectrum.Proj.stalkMap_toSpec, AlgebraicGeometry.ProjectiveSpectrum.Proj.awayToSection_germ
toCommSemiRingCatIso 📖	CompOp	2 math math: toCommSemiRingCatIso_inv, toCommSemiRingCatIso_hom
toRingCatIso 📖	CompOp	2 math math: toRingCatIso_inv, toRingCatIso_hom
toSemiRingCatIso 📖	CompOp	2 math math: toSemiRingCatIso_hom, toSemiRingCatIso_inv

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
toCommRingCatIso_hom 📖	mathematical	—	CategoryTheory.Iso.hom CommRingCat CommRingCat.instCategory CommRingCat.of toCommRingCatIso CommRingCat.ofHom RingHomClass.toRingHom RingEquiv Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing CommRing.toNonUnitalCommRing Distrib.toAdd EquivLike.toFunLike instEquivLike Semiring.toNonAssocSemiring CommSemiring.toSemiring CommRing.toCommSemiring	—	—
toCommRingCatIso_inv 📖	mathematical	—	CategoryTheory.Iso.inv CommRingCat CommRingCat.instCategory CommRingCat.of toCommRingCatIso CommRingCat.ofHom RingHomClass.toRingHom RingEquiv Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing CommRing.toNonUnitalCommRing Distrib.toAdd EquivLike.toFunLike instEquivLike Semiring.toNonAssocSemiring CommSemiring.toSemiring CommRing.toCommSemiring symm	—	—
toCommSemiRingCatIso_hom 📖	mathematical	—	CategoryTheory.Iso.hom CommSemiRingCat CommSemiRingCat.instCategory CommSemiRingCat.of toCommSemiRingCatIso CommSemiRingCat.ofHom RingHomClass.toRingHom RingEquiv Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonAssocSemiring.toNonUnitalNonAssocSemiring Semiring.toNonAssocSemiring CommSemiring.toSemiring Distrib.toAdd EquivLike.toFunLike instEquivLike	—	—
toCommSemiRingCatIso_inv 📖	mathematical	—	CategoryTheory.Iso.inv CommSemiRingCat CommSemiRingCat.instCategory CommSemiRingCat.of toCommSemiRingCatIso CommSemiRingCat.ofHom RingHomClass.toRingHom RingEquiv Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonAssocSemiring.toNonUnitalNonAssocSemiring Semiring.toNonAssocSemiring CommSemiring.toSemiring Distrib.toAdd EquivLike.toFunLike instEquivLike symm	—	—
toRingCatIso_hom 📖	mathematical	—	CategoryTheory.Iso.hom RingCat RingCat.instCategory RingCat.of toRingCatIso RingCat.ofHom RingHomClass.toRingHom RingEquiv Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonAssocRing.toNonUnitalNonAssocRing Ring.toNonAssocRing Distrib.toAdd EquivLike.toFunLike instEquivLike Semiring.toNonAssocSemiring Ring.toSemiring	—	—
toRingCatIso_inv 📖	mathematical	—	CategoryTheory.Iso.inv RingCat RingCat.instCategory RingCat.of toRingCatIso RingCat.ofHom RingHomClass.toRingHom RingEquiv Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonAssocRing.toNonUnitalNonAssocRing Ring.toNonAssocRing Distrib.toAdd EquivLike.toFunLike instEquivLike Semiring.toNonAssocSemiring Ring.toSemiring symm	—	—
toSemiRingCatIso_hom 📖	mathematical	—	CategoryTheory.Iso.hom SemiRingCat SemiRingCat.instCategory SemiRingCat.of toSemiRingCatIso SemiRingCat.ofHom RingHomClass.toRingHom RingEquiv Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonAssocSemiring.toNonUnitalNonAssocSemiring Semiring.toNonAssocSemiring Distrib.toAdd EquivLike.toFunLike instEquivLike	—	—
toSemiRingCatIso_inv 📖	mathematical	—	CategoryTheory.Iso.inv SemiRingCat SemiRingCat.instCategory SemiRingCat.of toSemiRingCatIso SemiRingCat.ofHom RingHomClass.toRingHom RingEquiv Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonAssocSemiring.toNonUnitalNonAssocSemiring Semiring.toNonAssocSemiring Distrib.toAdd EquivLike.toFunLike instEquivLike symm	—	—

SemiRingCat

Definitions

Name	Category	Theorems
carrier 📖	CompOp	36 math math: inv_hom_apply, comp_apply, ofHom_hom, hom_comp, RingCat.forget₂_map, FilteredColimits.colimitCoconeIsColimit.descMonoidHom_quotMk, forget₂Mon_preservesLimitsOfSize, hom_id, RingCat.forget₂SemiRing_preservesLimitsOfSize, forget_preservesLimitsOfSize, FilteredColimits.colimitCoconeIsColimit.descAddMonoidHom_quotMk, FilteredColimits.forget₂Mon_preservesFilteredColimits, CommSemiRingCat.forget₂SemiRing_preservesLimits, coe_of, FilteredColimits.forget_preservesFilteredColimits, forgetReflectIsos, FilteredColimits.colimitCoconeIsColimit.descMonoidHom_apply_eq, ofHom_apply, hom_inv_apply, CommSemiRingCat.FilteredColimits.forget₂SemiRing_preservesFilteredColimits, RingCat.instFullSemiRingCatForget₂RingHomCarrierCarrier, RingCat.forget₂SemiRing_preservesLimits, forget₂AddCommMon_preservesLimits, CommSemiRingCat.forget₂SemiRing_preservesLimitsOfSize, forget_preservesLimits, forget_obj, CategoryTheory.Iso.semiRingCatIsoToRingEquiv_toRingHom, id_apply, forget₂AddCommMon_preservesLimitsOfSize, forget₂_monCat_map, RingCat.FilteredColimits.forget₂SemiRing_preservesFilteredColimits, forget₂Mon_preservesLimits, forget_map, CommSemiRingCat.instFullSemiRingCatForget₂RingHomCarrierCarrier, forget₂_addCommMonCat_map, of_carrier
forget_obj_eq_coe 📖	MathDef	—
hasForgetToAddCommMonCat 📖	CompOp	3 math math: forget₂AddCommMon_preservesLimits, forget₂AddCommMon_preservesLimitsOfSize, forget₂_addCommMonCat_map
hasForgetToMonCat 📖	CompOp	6 math math: FilteredColimits.colimitCoconeIsColimit.descMonoidHom_quotMk, forget₂Mon_preservesLimitsOfSize, FilteredColimits.colimitCoconeIsColimit.descAddMonoidHom_quotMk, FilteredColimits.forget₂Mon_preservesFilteredColimits, forget₂_monCat_map, forget₂Mon_preservesLimits
instCategory 📖	CompOp	41 math math: hasLimit, inv_hom_apply, comp_apply, hom_comp, RingCat.forget₂_map, FilteredColimits.colimitCoconeIsColimit.descMonoidHom_quotMk, forget₂Mon_preservesLimitsOfSize, hom_id, RingCat.forget₂SemiRing_preservesLimitsOfSize, forget_preservesLimitsOfSize, FilteredColimits.colimitCoconeIsColimit.descAddMonoidHom_quotMk, FilteredColimits.forget₂Mon_preservesFilteredColimits, CommSemiRingCat.forget₂SemiRing_preservesLimits, FilteredColimits.forget_preservesFilteredColimits, hasLimitsOfShape, forgetReflectIsos, FilteredColimits.colimitCoconeIsColimit.descMonoidHom_apply_eq, hasLimits, ofHom_apply, hom_inv_apply, ofHom_id, CommSemiRingCat.FilteredColimits.forget₂SemiRing_preservesFilteredColimits, RingCat.instFullSemiRingCatForget₂RingHomCarrierCarrier, RingEquiv.toSemiRingCatIso_hom, RingCat.forget₂SemiRing_preservesLimits, forget₂AddCommMon_preservesLimits, CommSemiRingCat.forget₂SemiRing_preservesLimitsOfSize, forget_preservesLimits, forget_obj, CategoryTheory.Iso.semiRingCatIsoToRingEquiv_toRingHom, id_apply, hasLimitsOfSize, forget₂AddCommMon_preservesLimitsOfSize, forget₂_monCat_map, RingCat.FilteredColimits.forget₂SemiRing_preservesFilteredColimits, ofHom_comp, forget₂Mon_preservesLimits, forget_map, CommSemiRingCat.instFullSemiRingCatForget₂RingHomCarrierCarrier, RingEquiv.toSemiRingCatIso_inv, forget₂_addCommMonCat_map
instCoeSortType 📖	CompOp	—
instConcreteCategoryRingHomCarrier 📖	CompOp	29 math math: inv_hom_apply, comp_apply, RingCat.forget₂_map, FilteredColimits.colimitCoconeIsColimit.descMonoidHom_quotMk, forget₂Mon_preservesLimitsOfSize, RingCat.forget₂SemiRing_preservesLimitsOfSize, forget_preservesLimitsOfSize, FilteredColimits.colimitCoconeIsColimit.descAddMonoidHom_quotMk, FilteredColimits.forget₂Mon_preservesFilteredColimits, CommSemiRingCat.forget₂SemiRing_preservesLimits, FilteredColimits.forget_preservesFilteredColimits, forgetReflectIsos, ofHom_apply, hom_inv_apply, CommSemiRingCat.FilteredColimits.forget₂SemiRing_preservesFilteredColimits, RingCat.instFullSemiRingCatForget₂RingHomCarrierCarrier, RingCat.forget₂SemiRing_preservesLimits, forget₂AddCommMon_preservesLimits, CommSemiRingCat.forget₂SemiRing_preservesLimitsOfSize, forget_preservesLimits, forget_obj, id_apply, forget₂AddCommMon_preservesLimitsOfSize, forget₂_monCat_map, RingCat.FilteredColimits.forget₂SemiRing_preservesFilteredColimits, forget₂Mon_preservesLimits, forget_map, CommSemiRingCat.instFullSemiRingCatForget₂RingHomCarrierCarrier, forget₂_addCommMonCat_map
instInhabited 📖	CompOp	—
instSemiringObjForgetRingHomCarrier 📖	CompOp	—
ofHom 📖	CompOp	7 math math: ofHom_hom, ofHom_apply, ofHom_id, RingEquiv.toSemiRingCatIso_hom, hom_ofHom, ofHom_comp, RingEquiv.toSemiRingCatIso_inv
semiring 📖	CompOp	35 math math: inv_hom_apply, comp_apply, ofHom_hom, hom_comp, RingCat.forget₂_map, FilteredColimits.colimitCoconeIsColimit.descMonoidHom_quotMk, forget₂Mon_preservesLimitsOfSize, hom_id, RingCat.forget₂SemiRing_preservesLimitsOfSize, forget_preservesLimitsOfSize, FilteredColimits.colimitCoconeIsColimit.descAddMonoidHom_quotMk, FilteredColimits.forget₂Mon_preservesFilteredColimits, CommSemiRingCat.forget₂SemiRing_preservesLimits, FilteredColimits.forget_preservesFilteredColimits, forgetReflectIsos, FilteredColimits.colimitCoconeIsColimit.descMonoidHom_apply_eq, ofHom_apply, hom_inv_apply, CommSemiRingCat.FilteredColimits.forget₂SemiRing_preservesFilteredColimits, RingCat.instFullSemiRingCatForget₂RingHomCarrierCarrier, RingCat.forget₂SemiRing_preservesLimits, forget₂AddCommMon_preservesLimits, CommSemiRingCat.forget₂SemiRing_preservesLimitsOfSize, forget_preservesLimits, forget_obj, CategoryTheory.Iso.semiRingCatIsoToRingEquiv_toRingHom, id_apply, forget₂AddCommMon_preservesLimitsOfSize, forget₂_monCat_map, RingCat.FilteredColimits.forget₂SemiRing_preservesFilteredColimits, forget₂Mon_preservesLimits, forget_map, CommSemiRingCat.instFullSemiRingCatForget₂RingHomCarrierCarrier, forget₂_addCommMonCat_map, of_carrier

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
coe_of 📖	mathematical	—	carrier of	—	—
comp_apply 📖	mathematical	—	DFunLike.coe carrier CategoryTheory.ConcreteCategory.hom SemiRingCat instCategory RingHom Semiring.toNonAssocSemiring semiring RingHom.instFunLike instConcreteCategoryRingHomCarrier CategoryTheory.CategoryStruct.comp CategoryTheory.Category.toCategoryStruct	—	—
forgetReflectIsos 📖	mathematical	—	CategoryTheory.Functor.ReflectsIsomorphisms SemiRingCat instCategory CategoryTheory.types CategoryTheory.forget RingHom carrier Semiring.toNonAssocSemiring semiring RingHom.instFunLike instConcreteCategoryRingHomCarrier	—	Equiv.left_inv Equiv.right_inv MonoidHom.map_mul' RingHom.map_add' CategoryTheory.Iso.isIso_hom
forget_map 📖	mathematical	—	CategoryTheory.Functor.map SemiRingCat instCategory CategoryTheory.types CategoryTheory.forget RingHom carrier Semiring.toNonAssocSemiring semiring RingHom.instFunLike instConcreteCategoryRingHomCarrier DFunLike.coe CategoryTheory.ConcreteCategory.hom	—	—
forget_obj 📖	mathematical	—	CategoryTheory.Functor.obj SemiRingCat instCategory CategoryTheory.types CategoryTheory.forget RingHom carrier Semiring.toNonAssocSemiring semiring RingHom.instFunLike instConcreteCategoryRingHomCarrier	—	—
forget₂_addCommMonCat_map 📖	mathematical	—	DFunLike.coe CategoryTheory.Functor.obj SemiRingCat instCategory AddCommMonCat AddCommMonCat.instCategory CategoryTheory.forget₂ RingHom carrier Semiring.toNonAssocSemiring semiring RingHom.instFunLike instConcreteCategoryRingHomCarrier AddMonoidHom AddCommMonCat.carrier AddZeroClass.toAddZero AddMonoid.toAddZeroClass AddCommMonoid.toAddMonoid AddCommMonCat.str AddMonoidHom.instFunLike AddCommMonCat.instConcreteCategoryAddMonoidHomCarrier hasForgetToAddCommMonCat CategoryTheory.ConcreteCategory.hom CategoryTheory.Functor.map	—	—
forget₂_monCat_map 📖	mathematical	—	DFunLike.coe CategoryTheory.Functor.obj SemiRingCat instCategory MonCat MonCat.instCategory CategoryTheory.forget₂ RingHom carrier Semiring.toNonAssocSemiring semiring RingHom.instFunLike instConcreteCategoryRingHomCarrier MonoidHom MonCat.carrier MulOneClass.toMulOne Monoid.toMulOneClass MonCat.str MonoidHom.instFunLike MonCat.instConcreteCategoryMonoidHomCarrier hasForgetToMonCat CategoryTheory.ConcreteCategory.hom CategoryTheory.Functor.map	—	—
hom_comp 📖	mathematical	—	Hom.hom CategoryTheory.CategoryStruct.comp SemiRingCat CategoryTheory.Category.toCategoryStruct instCategory RingHom.comp carrier Semiring.toNonAssocSemiring semiring	—	—
hom_ext 📖	—	Hom.hom	—	—	Hom.ext
hom_ext_iff 📖	mathematical	—	Hom.hom	—	hom_ext
hom_id 📖	mathematical	—	Hom.hom CategoryTheory.CategoryStruct.id SemiRingCat CategoryTheory.Category.toCategoryStruct instCategory RingHom.id carrier Semiring.toNonAssocSemiring semiring	—	—
hom_inv_apply 📖	mathematical	—	DFunLike.coe carrier CategoryTheory.ConcreteCategory.hom SemiRingCat instCategory RingHom Semiring.toNonAssocSemiring semiring RingHom.instFunLike instConcreteCategoryRingHomCarrier CategoryTheory.Iso.hom CategoryTheory.Iso.inv	—	CategoryTheory.Iso.inv_hom_id_apply
hom_ofHom 📖	mathematical	—	Hom.hom of ofHom	—	—
id_apply 📖	mathematical	—	DFunLike.coe carrier CategoryTheory.ConcreteCategory.hom SemiRingCat instCategory RingHom Semiring.toNonAssocSemiring semiring RingHom.instFunLike instConcreteCategoryRingHomCarrier CategoryTheory.CategoryStruct.id CategoryTheory.Category.toCategoryStruct	—	—
inv_hom_apply 📖	mathematical	—	DFunLike.coe carrier CategoryTheory.ConcreteCategory.hom SemiRingCat instCategory RingHom Semiring.toNonAssocSemiring semiring RingHom.instFunLike instConcreteCategoryRingHomCarrier CategoryTheory.Iso.inv CategoryTheory.Iso.hom	—	CategoryTheory.Iso.hom_inv_id_apply
ofHom_apply 📖	mathematical	—	DFunLike.coe of carrier CategoryTheory.ConcreteCategory.hom SemiRingCat instCategory RingHom Semiring.toNonAssocSemiring semiring RingHom.instFunLike instConcreteCategoryRingHomCarrier ofHom	—	—
ofHom_comp 📖	mathematical	—	ofHom RingHom.comp Semiring.toNonAssocSemiring CategoryTheory.CategoryStruct.comp SemiRingCat CategoryTheory.Category.toCategoryStruct instCategory of	—	—
ofHom_hom 📖	mathematical	—	ofHom carrier semiring Hom.hom	—	—
ofHom_id 📖	mathematical	—	ofHom RingHom.id Semiring.toNonAssocSemiring CategoryTheory.CategoryStruct.id SemiRingCat CategoryTheory.Category.toCategoryStruct instCategory of	—	—
of_carrier 📖	mathematical	—	of carrier semiring	—	—

SemiRingCat.Hom

Definitions

Name	Category	Theorems
hom 📖	CompOp	6 math math: SemiRingCat.ofHom_hom, SemiRingCat.hom_comp, SemiRingCat.hom_id, SemiRingCat.hom_ext_iff, SemiRingCat.hom_ofHom, CategoryTheory.Iso.semiRingCatIsoToRingEquiv_toRingHom
hom' 📖	CompOp	1 math math: ext_iff

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
ext 📖	—	hom'	—	—	—
ext_iff 📖	mathematical	—	hom'	—	ext

SemiRingCat.Hom.Simps

Definitions

Name	Category	Theorems
hom 📖	CompOp	—

(root)

Definitions

Name	Category	Theorems
CommSemiRingCat 📖	CompData	28 math math: CommSemiRingCat.forget_obj, CommSemiRingCat.hom_inv_apply, CommRingCat.forget₂CommSemiRing_preservesLimitsOfSize, CommSemiRingCat.hom_id, RingEquiv.toCommSemiRingCatIso_inv, CommSemiRingCat.ofHom_comp, CommSemiRingCat.hasLimitsOfSize, CommSemiRingCat.ofHom_apply, CommSemiRingCat.ofHom_id, CommSemiRingCat.hasLimit, CommSemiRingCat.forgetReflectIsos, CommSemiRingCat.forget₂SemiRing_preservesLimits, CommSemiRingCat.forget_preservesLimits, CommRingCat.forget₂CommSemiRing_preservesLimits, CommSemiRingCat.hom_comp, CommSemiRingCat.FilteredColimits.forget_preservesFilteredColimits, RingEquiv.toCommSemiRingCatIso_hom, CommSemiRingCat.comp_apply, CommSemiRingCat.id_apply, CommSemiRingCat.forget_map, CommSemiRingCat.FilteredColimits.forget₂SemiRing_preservesFilteredColimits, CommSemiRingCat.forget₂SemiRing_preservesLimitsOfSize, CommSemiRingCat.forget_preservesLimitsOfSize, CommSemiRingCat.hasLimits, CommSemiRingCat.inv_hom_apply, CommSemiRingCat.hasLimitsOfShape, CommSemiRingCat.instFullSemiRingCatForget₂RingHomCarrierCarrier, CategoryTheory.Iso.commSemiRingCatIsoToRingEquiv_toRingHom
RingCat 📖	CompData	225 math math: PresheafOfModules.Monoidal.tensorObj_obj, PresheafOfModules.instIsRightAdjointPushforwardCompFunctorOppositeRingCatWhiskerLeftOp, SheafOfModules.pushforward_assoc, CommRingCat.forget₂Ring_preservesLimitsOfSize, AlgebraicGeometry.StructureSheaf.instIsScalarTowerCarrierStalkCommRingCatStructurePresheafInCommRingCatCarrierAbPresheafOpensCarrierTopModuleStructurePresheaf, RingEquiv.toRingCatIso_inv, CommRingCat.instFullRingCatForget₂RingHomCarrierCarrier, AlgebraicGeometry.Scheme.Modules.instFaithfulPresheafOfModulesToPresheafOfModules, RingCat.FilteredColimits.instPreservesFilteredColimitsAddCommGrpCatForget₂RingHomCarrierAddMonoidHomCarrier, PresheafOfModules.pullback_id_comp, PresheafOfModules.instIsLocallySurjectiveToSheafify, PresheafOfModules.add_app, SheafOfModules.pushforwardNatIso_inv, PresheafOfModules.Sheafify.add_smul, PresheafOfModules.restrictScalars_map_app, SheafOfModules.instFullPresheafOfModulesValRingCatForget, PresheafOfModules.epi_iff_surjective, PresheafOfModules.pullback_comp_id, PresheafOfModules.evaluation_preservesColimitsOfShape, PresheafOfModules.comp_app, RingCat.Colimits.cocone_naturality_components, PresheafOfModules.pushforward_map_app_apply, PresheafOfModules.sections_property, RingCat.forget₂AddCommGroup_preservesLimits, PresheafOfModules.toSheafify_app_apply', PresheafOfModules.instPreservesLimitsOfShapeModuleCatCarrierObjOppositeRingCatEvaluation, PresheafOfModules.toFreeYonedaCoproduct_fromFreeYonedaCoproduct, PresheafOfModules.Derivation.d_map, PresheafOfModules.free_map_app, RingCat.forget₂_map, SheafOfModules.instIsRightAdjointPushforwardCompSheafRingCatMapSheafPushforwardContinuous, PresheafOfModules.congr_map_apply, RingCat.hom_comp, PresheafOfModules.freeYonedaEquiv_symm_app, PresheafOfModules.restrictScalarsObj_map, PresheafOfModules.forgetToPresheafModuleCatObj_map, RingCat.inv_hom_apply, SheafOfModules.evaluationPreservesLimitsOfShape, SheafOfModules.toSheaf_obj_val, PresheafOfModules.pushforward₀_obj_obj_carrier, PresheafOfModules.id_app, PresheafOfModules.toPresheaf_map_toSheafify, PresheafOfModules.restrictScalarsObj_obj, PresheafOfModules.Derivation.postcomp_d_apply, RingCat.forget_map_apply, PresheafOfModules.Derivation.d_one, PresheafOfModules.sectionsMap_coe, PresheafOfModules.map_comp_apply, PresheafOfModules.pushforward_map_app_apply', PresheafOfModules.Derivation.d_mul, SheafOfModules.evaluationPreservesLimitsOfSize, PresheafOfModules.unit_map_one, PresheafOfModules.zsmul_app, PresheafOfModules.toPresheaf_obj_coe, AlgCat.forget₂Ring_preservesLimitsOfSize, PresheafOfModules.isoMk_hom_app, SheafOfModules.isSheaf, PresheafOfModules.map_smul, PresheafOfModules.Monoidal.tensorObj_map_tmul, RingCat.hasLimitsOfShape, PresheafOfModules.instPreservesLimitsOfSizeModuleCatCarrierObjOppositeRingCatEvaluation, SheafOfModules.pushforwardComp_inv_app_val_app, RingCat.forget₂SemiRing_preservesLimitsOfSize, PresheafOfModules.pushforward₀_obj_obj_isAddCommGroup, RingCat.Colimits.cocone_naturality, SheafOfModules.pushforwardNatTrans_id, SheafOfModules.restrictScalars_obj_val, PresheafOfModules.mono_iff_surjective, PresheafOfModules.freeObj_map, SheafOfModules.instReflectsIsomorphismsPresheafOfModulesValRingCatForget, PresheafOfModules.instEpiModuleCatCarrierObjOppositeRingCatApp, PresheafOfModules.presheaf_map_apply_coe, SheafOfModules.instFaithfulPresheafOfModulesValRingCatForget, PresheafOfModules.Derivation.congr_d, RingEquiv.toRingCatIso_hom, PresheafOfModules.forgetToPresheafModuleCatObjMap_apply, SheafOfModules.pushforwardCongr_hom_app_val_app, SheafOfModules.toSheaf_map_val, PresheafOfModules.pullback_assoc, RingCat.FilteredColimits.forget_preservesFilteredColimits, CategoryTheory.Iso.ringCatIsoToRingEquiv_toRingHom, PresheafOfModules.toFreeYonedaCoproduct_fromFreeYonedaCoproduct_assoc, SheafOfModules.pullback_comp_id, PresheafOfModules.surjective_of_epi, PresheafOfModules.toSheaf_map_sheafificationHomEquiv_symm, PresheafOfModules.injective_of_mono, PresheafOfModules.isoMk_inv_app, SheafOfModules.id_val, PresheafOfModules.pushforward_obj_map_apply, RingCat.forget_preservesLimitsOfSize, PresheafOfModules.forgetToPresheafModuleCatObjObj_coe, RingCat.hasLimits, RingCat.ofHom_apply, PresheafOfModules.unitHomEquiv_apply_coe, PresheafOfModules.freeObj_obj, SheafOfModules.pushforwardCongr_inv_app_val_app, CommRingCat.forgetToRingCat_obj, SheafOfModules.conjugateEquiv_pullbackComp_inv, RingCat.moduleCatRestrictScalarsPseudofunctor_mapId, PresheafOfModules.freeAdjunctionUnit_app, PresheafOfModules.presheaf_obj_coe, SheafOfModules.fullyFaithfulForget_preimage_val, SheafOfModules.pullback_id_comp, SheafOfModules.restrictScalars_map_val, RingCat.Colimits.hasColimits_ringCat, PresheafOfModules.toPresheaf_map_app_apply, RingCat.hasLimitsOfSize, PresheafOfModules.Derivation'.d_app, PresheafOfModules.neg_app, PresheafOfModules.map_comp, RingCat.moduleCatRestrictScalarsPseudofunctor_map, SheafOfModules.pushforward_map_val, PresheafOfModules.evaluation_preservesColimitsOfSize, SheafOfModules.pushforwardComp_hom_app_val_app, SheafOfModules.unit_val, PresheafOfModules.evaluation_preservesFiniteLimits, SheafOfModules.Presentation.map_relations_I, SheafOfModules.instIsLeftAdjointOverOverRingCatPushforwardIdSheafOver, PresheafOfModules.sub_app, PresheafOfModules.DifferentialsConstruction.relativeDifferentials'_map_d, PresheafOfModules.Elements.fromFreeYoneda_app_apply, PresheafOfModules.pushforward_obj_obj, SheafOfModules.Presentation.mapRelations_mapGenerators, SheafOfModules.relationsOfIsCokernelFree_I, CommRingCat.forget₂Ring_preservesLimits, RingCat.forgetReflectIsos, SheafOfModules.pushforwardCongr_symm, PresheafOfModules.instAdditiveModuleCatCarrierObjOppositeRingCatEvaluation, PresheafOfModules.Hom.naturality_assoc, RingCat.ofHom_id, SheafOfModules.add_val, PresheafOfModules.sheafifyMap_val, CommRingCat.FilteredColimits.forget₂Ring_preservesFilteredColimits, PresheafOfModules.restriction_app, PresheafOfModules.ι_fromFreeYonedaCoproduct_apply, PresheafOfModules.DifferentialsConstruction.relativeDifferentials'_obj, PresheafOfModules.Derivation.d_app, PresheafOfModules.forgetToPresheafModuleCat_obj, SheafOfModules.unitToPushforwardObjUnit_val_app_apply, AlgebraicGeometry.instIsIsoModulesSpecOfCarrierFromTildeΓUnitOpensCarrierCarrierCommRingCatRingCatSheaf, PresheafOfModules.forgetToPresheafModuleCatObj_obj, SheafOfModules.pullback_assoc, SheafOfModules.forget_map, PresheafOfModules.Monoidal.tensorHom_app, RingCat.instFullSemiRingCatForget₂RingHomCarrierCarrier, PresheafOfModules.DifferentialsConstruction.relativeDifferentials'_map, SheafOfModules.forgetPreservesLimitsOfSize, RingCat.moduleCatRestrictScalarsPseudofunctor_obj, PresheafOfModules.instMonoModuleCatCarrierObjOppositeRingCatApp, PresheafOfModules.Derivation'.app_apply, AlgebraicGeometry.StructureSheaf.instIsLocalizedModuleCarrierStalkAbPresheafOpensCarrierTopModuleStructurePresheafPrimeComplAsIdealToStalkₗ, SheafOfModules.Finite.forgetPreservesFiniteLimits, SheafOfModules.Presentation.mapRelations_mapGenerators_assoc, SheafOfModules.forget_obj, RingCat.hom_id, RingCat.moduleCatRestrictScalarsPseudofunctor_mapComp, SheafOfModules.comp_val_assoc, RingCat.forget₂SemiRing_preservesLimits, RingCat.ofHom_comp, RingCat.comp_apply, SheafOfModules.pushforwardNatTrans_app_val_app, PresheafOfModules.Sheafify.one_smul, RingCat.forget₂AddCommGroup_preservesLimitsOfSize, PresheafOfModules.pushforward_assoc, PresheafOfModules.map_id, AlgebraicGeometry.Scheme.Modules.instFullPresheafOfModulesToPresheafOfModules, SheafOfModules.pushforward_comp_id, SheafOfModules.relationsOfIsCokernelFree_s, PresheafOfModules.pushforward_obj_map_apply', PresheafOfModules.Hom.naturality, AlgCat.forget₂Ring_preservesLimits, RingCat.forget_preservesLimits, SheafOfModules.forgetToSheafModuleCat_map_val, SheafOfModules.conjugateEquiv_pullbackId_hom, RingCat.hasLimit, PresheafOfModules.freeYonedaEquiv_comp, PresheafOfModules.Sheafify.zero_smul, PresheafOfModules.sheafificationAdjunction_homEquiv_apply, AlgebraicGeometry.instIsQuasicoherentOpensCarrierCarrierCommRingCatSpecTilde, SheafOfModules.forgetPreservesLimitsOfShape, SheafOfModules.Presentation.IsFinite.finite_relations, SheafOfModules.pushforwardNatTrans_app_val_app_apply, PresheafOfModules.Finite.evaluation_preservesFiniteColimits, SheafOfModules.comp_val, PresheafOfModules.toPresheaf_map_sheafificationAdjunction_unit_app, SheafOfModules.pushforward_id_comp, RingCat.hom_inv_apply, PresheafOfModules.instIsLocallyInjectiveToSheafify, CommRingCat.forgetToRingCat_map_hom, PresheafOfModules.germ_ringCat_smul, AlgebraicGeometry.Scheme.Modules.instIsRightAdjointPresheafOfModulesToPresheafOfModules, PresheafOfModules.naturality_apply, PresheafOfModules.toPresheaf_map_sheafificationHomEquiv_def, PresheafOfModules.instFullRestrictScalarsIdFunctorOppositeRingCat, SheafOfModules.pushforwardSections_coe, PresheafOfModules.fromFreeYonedaCoproduct_app_mk, PresheafOfModules.Sheafify.map_smul, PresheafOfModules.germ_smul, PresheafOfModules.toSheafify_app_apply, SheafOfModules.instIsRightAdjointPushforwardIdSheafRingCat, PresheafOfModules.pushforward₀_obj_obj_isModule, RingCat.id_apply, SheafOfModules.forgetToSheafModuleCat_obj_val, PresheafOfModules.evaluation_obj, SheafOfModules.pushforwardNatIso_hom, RingCat.forget_obj, PresheafOfModules.pushforward_id_comp, RingCat.FilteredColimits.forget₂SemiRing_preservesFilteredColimits, PresheafOfModules.evaluation_map, PresheafOfModules.freeObjDesc_app, SheafOfModules.pushforward_obj_val, PresheafOfModules.Sheafify.mul_smul, AlgebraicGeometry.instIsIsoModulesSpecOfCarrierFromTildeΓFreeOpensCarrierCarrierCommRingCat, PresheafOfModules.zero_app, PresheafOfModules.map_comp_assoc, PresheafOfModules.forgetToPresheafModuleCat_map, PresheafOfModules.toPresheaf_map_sheafificationHomEquiv, PresheafOfModules.instIsRightAdjointPushforwardIdFunctorOppositeRingCat, SheafOfModules.Finite.evaluationPreservesFiniteLimits, SheafOfModules.instAdditivePresheafOfModulesValRingCatForget, PresheafOfModules.pushforward_comp_id, SheafOfModules.pushforwardNatTrans_comp, RingCat.forget_map, PresheafOfModules.pushforward₀_obj_map, SheafOfModules.unitHomEquiv_apply_coe
SemiRingCat 📖	CompData	41 math math: SemiRingCat.hasLimit, SemiRingCat.inv_hom_apply, SemiRingCat.comp_apply, SemiRingCat.hom_comp, RingCat.forget₂_map, SemiRingCat.FilteredColimits.colimitCoconeIsColimit.descMonoidHom_quotMk, SemiRingCat.forget₂Mon_preservesLimitsOfSize, SemiRingCat.hom_id, RingCat.forget₂SemiRing_preservesLimitsOfSize, SemiRingCat.forget_preservesLimitsOfSize, SemiRingCat.FilteredColimits.colimitCoconeIsColimit.descAddMonoidHom_quotMk, SemiRingCat.FilteredColimits.forget₂Mon_preservesFilteredColimits, CommSemiRingCat.forget₂SemiRing_preservesLimits, SemiRingCat.FilteredColimits.forget_preservesFilteredColimits, SemiRingCat.hasLimitsOfShape, SemiRingCat.forgetReflectIsos, SemiRingCat.FilteredColimits.colimitCoconeIsColimit.descMonoidHom_apply_eq, SemiRingCat.hasLimits, SemiRingCat.ofHom_apply, SemiRingCat.hom_inv_apply, SemiRingCat.ofHom_id, CommSemiRingCat.FilteredColimits.forget₂SemiRing_preservesFilteredColimits, RingCat.instFullSemiRingCatForget₂RingHomCarrierCarrier, RingEquiv.toSemiRingCatIso_hom, RingCat.forget₂SemiRing_preservesLimits, SemiRingCat.forget₂AddCommMon_preservesLimits, CommSemiRingCat.forget₂SemiRing_preservesLimitsOfSize, SemiRingCat.forget_preservesLimits, SemiRingCat.forget_obj, CategoryTheory.Iso.semiRingCatIsoToRingEquiv_toRingHom, SemiRingCat.id_apply, SemiRingCat.hasLimitsOfSize, SemiRingCat.forget₂AddCommMon_preservesLimitsOfSize, SemiRingCat.forget₂_monCat_map, RingCat.FilteredColimits.forget₂SemiRing_preservesFilteredColimits, SemiRingCat.ofHom_comp, SemiRingCat.forget₂Mon_preservesLimits, SemiRingCat.forget_map, CommSemiRingCat.instFullSemiRingCatForget₂RingHomCarrierCarrier, RingEquiv.toSemiRingCatIso_inv, SemiRingCat.forget₂_addCommMonCat_map

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