Documentation

Mathlib.RingTheory.DedekindDomain.AdicValuation

Adic valuations on Dedekind domains #

Given a Dedekind domain R of Krull dimension 1 and a maximal ideal v of R, we define the v-adic valuation on R and its extension to the field of fractions K of R. We prove several properties of this valuation, including the existence of uniformizers.

We define the completion of K with respect to the v-adic valuation, denoted v.adicCompletion, and its ring of integers, denoted v.adicCompletionIntegers.

Main definitions #

IsDedekindDomain.HeightOneSpectrum.intValuation v is the v-adic valuation on R.
IsDedekindDomain.HeightOneSpectrum.valuation v is the v-adic valuation on K.
IsDedekindDomain.HeightOneSpectrum.adicCompletion v is the completion of K with respect to its v-adic valuation.
IsDedekindDomain.HeightOneSpectrum.adicCompletionIntegers v is the ring of integers of v.adicCompletion.
IsDedekindDomain.HeightOneSpectrum.adicAbv v is the v-adic absolute value on K defined as b raised to negative v-adic valuation, for some b in ℝ≥0.

Main results #

IsDedekindDomain.HeightOneSpectrum.intValuation_le_one : The v-adic valuation on R is bounded above by 1.
IsDedekindDomain.HeightOneSpectrum.intValuation_lt_one_iff_dvd : The v-adic valuation of r ∈ R is less than 1 if and only if v divides the ideal (r).
IsDedekindDomain.HeightOneSpectrum.intValuation_le_pow_iff_dvd : The v-adic valuation of r ∈ R is less than or equal to WithZero.exp (-n) if and only if vⁿ divides the ideal (r).
IsDedekindDomain.HeightOneSpectrum.intValuation_exists_uniformizer : There exists π ∈ R with v-adic valuation WithZero.exp (-1).
IsDedekindDomain.HeightOneSpectrum.valuation_of_mk' : The v-adic valuation of r/s ∈ K is the valuation of r divided by the valuation of s.
IsDedekindDomain.HeightOneSpectrum.valuation_of_algebraMap : The v-adic valuation on K extends the v-adic valuation on R.
IsDedekindDomain.HeightOneSpectrum.valuation_exists_uniformizer : There exists π ∈ K with v-adic valuation WithZero.exp (-1).

Implementation notes #

We are only interested in Dedekind domains with Krull dimension 1.

References #

[G. J. Janusz, Algebraic Number Fields][janusz1996]
[J.W.S. Cassels, A. Fröhlich, Algebraic Number Theory][cassels1967algebraic]
[J. Neukirch, Algebraic Number Theory][Neukirch1992]

Tags #

dedekind domain, dedekind ring, adic valuation

Adic valuations on the Dedekind domain R #

def IsDedekindDomain.HeightOneSpectrum.intValuationDef {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : HeightOneSpectrum R) (r : R) :

WithZero (Multiplicative ℤ)

The additive v-adic valuation of r ∈ R is the exponent of v in the factorization of the ideal (r), if r is nonzero, or infinity, if r = 0. intValuationDef is the corresponding multiplicative valuation.

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theorem IsDedekindDomain.HeightOneSpectrum.intValuationDef_if_pos {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : HeightOneSpectrum R) {r : R} (hr : r = 0) :

v.intValuationDef r = 0

@[simp]

theorem IsDedekindDomain.HeightOneSpectrum.intValuationDef_zero {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : HeightOneSpectrum R) :

v.intValuationDef 0 = 0

theorem IsDedekindDomain.HeightOneSpectrum.intValuationDef_if_neg {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : HeightOneSpectrum R) {r : R} (hr : r ≠ 0) :

v.intValuationDef r = WithZero.exp (-↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {r})).factors))

theorem IsDedekindDomain.HeightOneSpectrum.intValuation.map_zero' {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : HeightOneSpectrum R) :

v.intValuationDef 0 = 0

The v-adic valuation of 0 : R equals 0.

theorem IsDedekindDomain.HeightOneSpectrum.intValuation.map_one' {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : HeightOneSpectrum R) :

v.intValuationDef 1 = 1

The v-adic valuation of 1 : R equals 1.

theorem IsDedekindDomain.HeightOneSpectrum.intValuation.map_mul' {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : HeightOneSpectrum R) (x y : R) :

v.intValuationDef (x * y) = v.intValuationDef x * v.intValuationDef y

The v-adic valuation of a product equals the product of the valuations.

theorem IsDedekindDomain.HeightOneSpectrum.intValuation.le_max_iff_min_le {a b c : ℕ} :

Multiplicative.ofAdd (-↑c) ≤ max (Multiplicative.ofAdd (-↑a)) (Multiplicative.ofAdd (-↑b)) ↔ min a b ≤ c

theorem IsDedekindDomain.HeightOneSpectrum.intValuation.map_add_le_max' {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : HeightOneSpectrum R) (x y : R) :

v.intValuationDef (x + y) ≤ max (v.intValuationDef x) (v.intValuationDef y)

The v-adic valuation of a sum is bounded above by the maximum of the valuations.

def IsDedekindDomain.HeightOneSpectrum.intValuation {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : HeightOneSpectrum R) :

Valuation R (WithZero (Multiplicative ℤ))

The v-adic valuation on R.

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theorem IsDedekindDomain.HeightOneSpectrum.intValuation_apply {R : Type u_1} [CommRing R] [IsDedekindDomain R] {r : R} (v : HeightOneSpectrum R) :

v.intValuation r = v.intValuationDef r

theorem IsDedekindDomain.HeightOneSpectrum.intValuation_def {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : HeightOneSpectrum R) {r : R} :

v.intValuation r = if r = 0 then 0 else WithZero.exp (-↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {r})).factors))

theorem IsDedekindDomain.HeightOneSpectrum.intValuation_if_neg {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : HeightOneSpectrum R) {r : R} (hr : r ≠ 0) :

v.intValuation r = WithZero.exp (-↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {r})).factors))

theorem IsDedekindDomain.HeightOneSpectrum.intValuation_ne_zero {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : HeightOneSpectrum R) (x : R) (hx : x ≠ 0) :

v.intValuation x ≠ 0

Nonzero elements have nonzero adic valuation.

theorem IsDedekindDomain.HeightOneSpectrum.intValuation_ne_zero' {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : HeightOneSpectrum R) (x : ↥(nonZeroDivisors R)) :

v.intValuation ↑x ≠ 0

Nonzero divisors have nonzero valuation.

theorem IsDedekindDomain.HeightOneSpectrum.intValuation_zero_lt {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : HeightOneSpectrum R) (x : ↥(nonZeroDivisors R)) :

0 < v.intValuation ↑x

Nonzero divisors have valuation greater than zero.

theorem IsDedekindDomain.HeightOneSpectrum.intValuation_le_one {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : HeightOneSpectrum R) (x : R) :

v.intValuation x ≤ 1

The v-adic valuation on R is bounded above by 1.

theorem IsDedekindDomain.HeightOneSpectrum.intValuation_lt_one_iff_dvd {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : HeightOneSpectrum R) (r : R) :

v.intValuation r < 1 ↔ v.asIdeal ∣ Ideal.span {r}

The v-adic valuation of r ∈ R is less than 1 if and only if v divides the ideal (r).

theorem IsDedekindDomain.HeightOneSpectrum.intValuation_lt_one_iff_mem {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : HeightOneSpectrum R) (r : R) :

v.intValuation r < 1 ↔ r ∈ v.asIdeal

The v-adic valuation of r ∈ R is less than 1 if and only if r ∈ v.

theorem IsDedekindDomain.HeightOneSpectrum.intValuation_le_pow_iff_dvd {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : HeightOneSpectrum R) (r : R) (n : ℕ) :

v.intValuation r ≤ WithZero.exp (-↑n) ↔ v.asIdeal ^ n ∣ Ideal.span {r}

The v-adic valuation of r ∈ R is less than WithZero.exp (-n) if and only if vⁿ divides the ideal (r).

theorem IsDedekindDomain.HeightOneSpectrum.intValuation_le_pow_iff_mem {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : HeightOneSpectrum R) (r : R) (n : ℕ) :

v.intValuation r ≤ WithZero.exp (-↑n) ↔ r ∈ v.asIdeal ^ n

The v-adic valuation of r ∈ R is less than WithZero.exp (-n) if and only if r ∈ vⁿ.

theorem IsDedekindDomain.HeightOneSpectrum.intValuation_exists_uniformizer {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : HeightOneSpectrum R) :

∃ (π : R), v.intValuation π = WithZero.exp (-1)

There exists π ∈ R with v-adic valuation WithZero.exp (-1).

theorem IsDedekindDomain.HeightOneSpectrum.intValuation_singleton {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : HeightOneSpectrum R) {r : R} (hr : r ≠ 0) (hv : v.asIdeal = Ideal.span {r}) :

v.intValuation r = WithZero.exp (-1)

The I-adic valuation of a generator of I equals (-1 : ℤᵐ⁰)

@[simp]

theorem IsDedekindDomain.HeightOneSpectrum.intValuation_eq_one_iff {R : Type u_1} [CommRing R] [IsDedekindDomain R] {v : HeightOneSpectrum R} {x : R} :

v.intValuation x = 1 ↔ x ∉ v.asIdeal

Adic valuations on the field of fractions `K` #

def IsDedekindDomain.HeightOneSpectrum.valuation {R : Type u_1} [CommRing R] [IsDedekindDomain R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] (v : HeightOneSpectrum R) :

Valuation K (WithZero (Multiplicative ℤ))

The v-adic valuation of x ∈ K is the valuation of r divided by the valuation of s, where r and s are chosen so that x = r/s.

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theorem IsDedekindDomain.HeightOneSpectrum.valuation_def {R : Type u_1} [CommRing R] [IsDedekindDomain R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] (v : HeightOneSpectrum R) (x : K) :

(valuation K v) x = (v.intValuation.extendToLocalization ⋯ K) x

theorem IsDedekindDomain.HeightOneSpectrum.valuation_of_mk' {R : Type u_1} [CommRing R] [IsDedekindDomain R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] (v : HeightOneSpectrum R) {r : R} {s : ↥(nonZeroDivisors R)} :

(valuation K v) (IsLocalization.mk' K r s) = v.intValuation r / v.intValuation ↑s

The v-adic valuation of r/s ∈ K is the valuation of r divided by the valuation of s.

theorem IsDedekindDomain.HeightOneSpectrum.valuation_of_algebraMap {R : Type u_1} [CommRing R] [IsDedekindDomain R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] (v : HeightOneSpectrum R) (r : R) :

(valuation K v) ↑r = v.intValuation r

The v-adic valuation on K extends the v-adic valuation on R.

theorem IsDedekindDomain.HeightOneSpectrum.valuation_le_one {R : Type u_1} [CommRing R] [IsDedekindDomain R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] (v : HeightOneSpectrum R) (r : R) :

(valuation K v) ↑r ≤ 1

The v-adic valuation on R is bounded above by 1.

theorem IsDedekindDomain.HeightOneSpectrum.valuation_lt_one_iff_dvd {R : Type u_1} [CommRing R] [IsDedekindDomain R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] (v : HeightOneSpectrum R) (r : R) :

(valuation K v) ↑r < 1 ↔ v.asIdeal ∣ Ideal.span {r}

The v-adic valuation of r ∈ R is less than 1 if and only if v divides the ideal (r).

theorem IsDedekindDomain.HeightOneSpectrum.valuation_lt_one_iff_mem {R : Type u_1} [CommRing R] [IsDedekindDomain R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] (v : HeightOneSpectrum R) (r : R) :

(valuation K v) ↑r < 1 ↔ r ∈ v.asIdeal

The v-adic valuation of r ∈ R is less than 1 if and only if r ∈ v.

theorem IsDedekindDomain.HeightOneSpectrum.valuation_div_le_one_iff {R : Type u_1} [CommRing R] [IsDedekindDomain R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] (v : HeightOneSpectrum R) (a : R) {b : R} (hb : b ≠ 0) (h : b ∈ v.asIdeal → a ∉ v.asIdeal) :

(valuation K v) (↑a / ↑b) ≤ 1 ↔ b ∉ v.asIdeal

The v adic valuation of a / b ∈ K is ≤ 1 if and only if b ∉ v, provided that a and b are coprime at v.

theorem IsDedekindDomain.HeightOneSpectrum.valuation_exists_uniformizer {R : Type u_1} [CommRing R] [IsDedekindDomain R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] (v : HeightOneSpectrum R) :

∃ (π : K), (valuation K v) π = WithZero.exp (-1)

There exists π ∈ K with v-adic valuation WithZero.exp (-1).

instance IsDedekindDomain.HeightOneSpectrum.instIsNontrivialWithZeroMultiplicativeIntValuation {R : Type u_1} [CommRing R] [IsDedekindDomain R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] (v : HeightOneSpectrum R) :

(valuation K v).IsNontrivial

theorem IsDedekindDomain.HeightOneSpectrum.valuation_surjective {R : Type u_1} [CommRing R] [IsDedekindDomain R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] (v : HeightOneSpectrum R) :

Function.Surjective ⇑(valuation K v)

theorem IsDedekindDomain.HeightOneSpectrum.valuation_uniformizer_ne_zero {R : Type u_1} [CommRing R] [IsDedekindDomain R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] (v : HeightOneSpectrum R) :

Classical.choose ⋯ ≠ 0

Uniformizers are nonzero.

theorem IsDedekindDomain.HeightOneSpectrum.mem_integers_of_valuation_le_one {R : Type u_1} [CommRing R] [IsDedekindDomain R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] (x : K) (h : ∀ (v : HeightOneSpectrum R), (valuation K v) x ≤ 1) :

x ∈ (algebraMap R K).range

theorem IsDedekindDomain.HeightOneSpectrum.eq_of_valuation_isEquiv_valuation {R : Type u_1} [CommRing R] [IsDedekindDomain R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] {p q : HeightOneSpectrum R} (hpq : (valuation K p).IsEquiv (valuation K q)) :

p = q

Completions with respect to adic valuations #

Given a Dedekind domain R with field of fractions K and a maximal ideal v of R, we define the completion of K with respect to its v-adic valuation, denoted v.adicCompletion, and its ring of integers, denoted v.adicCompletionIntegers.

def IsDedekindDomain.HeightOneSpectrum.adicValued {R : Type u_1} [CommRing R] [IsDedekindDomain R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] (v : HeightOneSpectrum R) :

Valued K (WithZero (Multiplicative ℤ))

K as a valued field with the v-adic valuation.

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theorem IsDedekindDomain.HeightOneSpectrum.adicValued_apply {R : Type u_1} [CommRing R] [IsDedekindDomain R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] (v : HeightOneSpectrum R) {x : K} :

Valued.v x = (valuation K v) x

@[simp]

theorem IsDedekindDomain.HeightOneSpectrum.adicValued_apply' {R : Type u_1} [CommRing R] [IsDedekindDomain R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] (v : HeightOneSpectrum R) (x : WithVal (valuation K v)) :

Valued.v x = (valuation K v) x

@[reducible, inline]

abbrev IsDedekindDomain.HeightOneSpectrum.adicCompletion {R : Type u_1} [CommRing R] [IsDedekindDomain R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] (v : HeightOneSpectrum R) :

Type u_2

The completion of K with respect to its v-adic valuation.

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theorem IsDedekindDomain.HeightOneSpectrum.valuedAdicCompletion_def {R : Type u_1} [CommRing R] [IsDedekindDomain R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] (v : HeightOneSpectrum R) {x : adicCompletion K v} :

Valued.v x = Valued.extension x

theorem IsDedekindDomain.HeightOneSpectrum.valuedAdicCompletion_surjective {R : Type u_1} [CommRing R] [IsDedekindDomain R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] (v : HeightOneSpectrum R) :

Function.Surjective ⇑Valued.v

def IsDedekindDomain.HeightOneSpectrum.adicCompletionIntegers {R : Type u_1} [CommRing R] [IsDedekindDomain R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] (v : HeightOneSpectrum R) :

ValuationSubring (adicCompletion K v)

The ring of integers of adicCompletion.

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instance IsDedekindDomain.HeightOneSpectrum.instInhabitedSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegers {R : Type u_1} [CommRing R] [IsDedekindDomain R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] (v : HeightOneSpectrum R) :

Inhabited ↥(adicCompletionIntegers K v)

Equations

theorem IsDedekindDomain.HeightOneSpectrum.mem_adicCompletionIntegers (R : Type u_1) [CommRing R] [IsDedekindDomain R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] (v : HeightOneSpectrum R) {x : adicCompletion K v} :

x ∈ adicCompletionIntegers K v ↔ Valued.v x ≤ 1

theorem IsDedekindDomain.HeightOneSpectrum.notMem_adicCompletionIntegers (R : Type u_1) [CommRing R] [IsDedekindDomain R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] (v : HeightOneSpectrum R) {x : adicCompletion K v} :

x ∉ adicCompletionIntegers K v ↔ 1 < Valued.v x

@[instance 100]

instance IsDedekindDomain.HeightOneSpectrum.adicValued.has_uniform_continuous_const_smul' (R : Type u_1) [CommRing R] [IsDedekindDomain R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] (v : HeightOneSpectrum R) :

UniformContinuousConstSMul R (WithVal (valuation K v))

instance IsDedekindDomain.HeightOneSpectrum.adicValued.uniformContinuousConstSMul (R : Type u_1) [CommRing R] [IsDedekindDomain R] (K : Type u_2) {S : Type u_3} [Field K] [CommSemiring S] [Algebra R K] [IsFractionRing R K] (v : HeightOneSpectrum R) [Algebra S K] :

UniformContinuousConstSMul S (WithVal (valuation K v))

instance IsDedekindDomain.HeightOneSpectrum.instAlgebraAdicCompletion (R : Type u_1) [CommRing R] [IsDedekindDomain R] (K : Type u_2) {S : Type u_3} [Field K] [CommSemiring S] [Algebra R K] [IsFractionRing R K] (v : HeightOneSpectrum R) [Algebra S K] :

Algebra S (adicCompletion K v)

Equations

theorem IsDedekindDomain.HeightOneSpectrum.coe_smul_adicCompletion (R : Type u_1) [CommRing R] [IsDedekindDomain R] (K : Type u_2) {S : Type u_3} [Field K] [CommSemiring S] [Algebra R K] [IsFractionRing R K] (v : HeightOneSpectrum R) [Algebra S K] (r : S) (x : WithVal (valuation K v)) :

↑(r • x) = r • ↑x

theorem IsDedekindDomain.HeightOneSpectrum.algebraMap_adicCompletion (R : Type u_1) [CommRing R] [IsDedekindDomain R] (K : Type u_2) {S : Type u_3} [Field K] [CommSemiring S] [Algebra R K] [IsFractionRing R K] (v : HeightOneSpectrum R) [Algebra S K] :

⇑(algebraMap S (adicCompletion K v)) = UniformSpace.Completion.coe' ∘ ⇑(algebraMap S K)

theorem IsDedekindDomain.HeightOneSpectrum.coe_algebraMap_mem (R : Type u_1) [CommRing R] [IsDedekindDomain R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] (v : HeightOneSpectrum R) (r : R) :

↑((algebraMap R K) r) ∈ adicCompletionIntegers K v

instance IsDedekindDomain.HeightOneSpectrum.instAlgebraSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegers (R : Type u_1) [CommRing R] [IsDedekindDomain R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] (v : HeightOneSpectrum R) :

Algebra R ↥(adicCompletionIntegers K v)

Equations

@[simp]

theorem IsDedekindDomain.HeightOneSpectrum.algebraMap_adicCompletionIntegers_apply (R : Type u_1) [CommRing R] [IsDedekindDomain R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] (v : HeightOneSpectrum R) (r : R) :

↑((algebraMap R ↥(adicCompletionIntegers K v)) r) = ↑((algebraMap R K) r)

instance IsDedekindDomain.HeightOneSpectrum.instFaithfulSMulSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegers (R : Type u_1) [CommRing R] [IsDedekindDomain R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] (v : HeightOneSpectrum R) [FaithfulSMul R K] :

FaithfulSMul R ↥(adicCompletionIntegers K v)

theorem IsDedekindDomain.HeightOneSpectrum.valuedAdicCompletion_eq_valuation {R : Type u_1} [CommRing R] [IsDedekindDomain R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] (v : HeightOneSpectrum R) (r : R) :

Valued.v ↑r = (valuation K v) ↑r

The valuation on the completion agrees with the global valuation on elements of the integer ring.

theorem IsDedekindDomain.HeightOneSpectrum.valuedAdicCompletion_eq_valuation' {R : Type u_1} [CommRing R] [IsDedekindDomain R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] (v : HeightOneSpectrum R) (k : K) :

Valued.v ↑k = (valuation K v) k

The valuation on the completion agrees with the global valuation on elements of the field.

theorem IsDedekindDomain.HeightOneSpectrum.coe_mem_adicCompletionIntegers {R : Type u_1} [CommRing R] [IsDedekindDomain R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] (v : HeightOneSpectrum R) (r : R) :

↑r ∈ adicCompletionIntegers K v

A global integer is in the local integers.

@[simp]

theorem IsDedekindDomain.HeightOneSpectrum.coe_smul_adicCompletionIntegers (R : Type u_1) [CommRing R] [IsDedekindDomain R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] (v : HeightOneSpectrum R) (r : R) (x : ↥(adicCompletionIntegers K v)) :

↑(r • x) = r • ↑x

instance IsDedekindDomain.HeightOneSpectrum.instIsTorsionFreeSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegers (R : Type u_1) [CommRing R] [IsDedekindDomain R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] (v : HeightOneSpectrum R) :

Module.IsTorsionFree R ↥(adicCompletionIntegers K v)

instance IsDedekindDomain.HeightOneSpectrum.adicCompletion.instIsScalarTower' (R : Type u_1) [CommRing R] [IsDedekindDomain R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] (v : HeightOneSpectrum R) :

IsScalarTower R (↥(adicCompletionIntegers K v)) (adicCompletion K v)

theorem IsDedekindDomain.HeightOneSpectrum.adicCompletion.mul_nonZeroDivisor_mem_adicCompletionIntegers {R : Type u_1} [CommRing R] [IsDedekindDomain R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] (v : HeightOneSpectrum R) (a : adicCompletion K v) :

∃ b ∈ nonZeroDivisors R, a * ↑b ∈ adicCompletionIntegers K v

theorem IsDedekindDomain.HeightOneSpectrum.adicCompletionIntegers.integers {R : Type u_1} [CommRing R] [IsDedekindDomain R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] (v : HeightOneSpectrum R) :

Valued.v.Integers ↥(adicCompletionIntegers K v)

theorem IsDedekindDomain.HeightOneSpectrum.adicCompletionIntegers.isUnit_iff_valued_eq_one {R : Type u_1} [CommRing R] [IsDedekindDomain R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] {v : HeightOneSpectrum R} {a : ↥(adicCompletionIntegers K v)} :

IsUnit a ↔ Valued.v ↑a = 1

theorem IsDedekindDomain.HeightOneSpectrum.adicCompletionIntegers.mem_units_iff_valued_eq_one {R : Type u_1} [CommRing R] [IsDedekindDomain R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] {v : HeightOneSpectrum R} {a : (adicCompletion K v)ˣ} :

a ∈ (adicCompletionIntegers K v).units ↔ Valued.v ↑a = 1

def IsDedekindDomain.HeightOneSpectrum.intAdicAbvDef {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : HeightOneSpectrum R) {b : NNReal} (hb : 1 < b) (r : R) :

The v-adic absolute value function on R defined as b raised to negative v-adic valuation, for some b in ℝ≥0

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theorem IsDedekindDomain.HeightOneSpectrum.isNonarchimedean_intAdicAbvDef {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : HeightOneSpectrum R) {b : NNReal} (hb : 1 < b) :

IsNonarchimedean (v.intAdicAbvDef hb)

def IsDedekindDomain.HeightOneSpectrum.intAdicAbv {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : HeightOneSpectrum R) {b : NNReal} (hb : 1 < b) :

AbsoluteValue R ℝ

The v-adic absolute value on R defined as b raised to negative v-adic valuation, for some b in ℝ≥0

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theorem IsDedekindDomain.HeightOneSpectrum.isNonarchimedean_intAdicAbv {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : HeightOneSpectrum R) {b : NNReal} (hb : 1 < b) :

IsNonarchimedean ⇑(v.intAdicAbv hb)

The v-adic absolute value is nonarchimedean

theorem IsDedekindDomain.HeightOneSpectrum.intAdicAbv_le_one {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : HeightOneSpectrum R) {b : NNReal} (hb : 1 < b) (r : R) :

(v.intAdicAbv hb) r ≤ 1

theorem IsDedekindDomain.HeightOneSpectrum.intAdicAbv_lt_one_iff {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : HeightOneSpectrum R) {b : NNReal} (hb : 1 < b) (r : R) :

(v.intAdicAbv hb) r < 1 ↔ r ∈ v.asIdeal

theorem IsDedekindDomain.HeightOneSpectrum.intAdicAbv_eq_one_iff {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : HeightOneSpectrum R) {b : NNReal} (hb : 1 < b) (r : R) :

(v.intAdicAbv hb) r = 1 ↔ r ∉ v.asIdeal

def IsDedekindDomain.HeightOneSpectrum.adicAbvDef {R : Type u_1} [CommRing R] [IsDedekindDomain R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] (v : HeightOneSpectrum R) {b : NNReal} (hb : 1 < b) (x : K) :

The v-adic absolute value function on K defined as b raised to negative v-adic valuation, for some b in ℝ≥0

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theorem IsDedekindDomain.HeightOneSpectrum.isNonarchimedean_adicAbvDef {R : Type u_1} [CommRing R] [IsDedekindDomain R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] (v : HeightOneSpectrum R) {b : NNReal} (hb : 1 < b) :

IsNonarchimedean (v.adicAbvDef hb)

def IsDedekindDomain.HeightOneSpectrum.adicAbv {R : Type u_1} [CommRing R] [IsDedekindDomain R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] (v : HeightOneSpectrum R) {b : NNReal} (hb : 1 < b) :

AbsoluteValue K ℝ

The v-adic absolute value on K defined as b raised to negative v-adic valuation, for some b in ℝ≥0

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theorem IsDedekindDomain.HeightOneSpectrum.isNonarchimedean_adicAbv {R : Type u_1} [CommRing R] [IsDedekindDomain R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] (v : HeightOneSpectrum R) {b : NNReal} (hb : 1 < b) :

IsNonarchimedean ⇑(v.adicAbv hb)

The v-adic absolute value is nonarchimedean

theorem IsDedekindDomain.HeightOneSpectrum.adicAbv_of_mk' {R : Type u_1} [CommRing R] [IsDedekindDomain R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] (v : HeightOneSpectrum R) {b : NNReal} (hb : 1 < b) (r : R) {s : ↥(nonZeroDivisors R)} :

(v.adicAbv hb) (IsLocalization.mk' K r s) = (v.intAdicAbv hb) r / (v.intAdicAbv hb) ↑s

The v-adic absolute value of r/s ∈ K is the absolute value of r divided by the absolute value of s.

theorem IsDedekindDomain.HeightOneSpectrum.adicAbv_of_algebraMap {R : Type u_1} [CommRing R] [IsDedekindDomain R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] (v : HeightOneSpectrum R) {b : NNReal} (hb : 1 < b) (r : R) :

(v.adicAbv hb) ((algebraMap R K) r) = (v.intAdicAbv hb) r

The v-adic absolute value on K extends the v-adic absolute value on R.

theorem IsDedekindDomain.HeightOneSpectrum.adicAbv_coe_le_one {R : Type u_1} [CommRing R] [IsDedekindDomain R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] (v : HeightOneSpectrum R) {b : NNReal} (hb : 1 < b) (r : R) :

(v.adicAbv hb) ((algebraMap R K) r) ≤ 1

theorem IsDedekindDomain.HeightOneSpectrum.adicAbv_coe_lt_one_iff {R : Type u_1} [CommRing R] [IsDedekindDomain R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] (v : HeightOneSpectrum R) {b : NNReal} (hb : 1 < b) (r : R) :

(v.adicAbv hb) ((algebraMap R K) r) < 1 ↔ r ∈ v.asIdeal

theorem IsDedekindDomain.HeightOneSpectrum.adicAbv_coe_eq_one_iff {R : Type u_1} [CommRing R] [IsDedekindDomain R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] (v : HeightOneSpectrum R) {b : NNReal} (hb : 1 < b) (r : R) :

(v.adicAbv hb) ((algebraMap R K) r) = 1 ↔ r ∉ v.asIdeal

theorem Rat.valuation_le_one_iff_den {R : Type u_4} [CommRing R] [IsDedekindDomain R] [Algebra R ℚ] [IsFractionRing R ℚ] {𝔭 : IsDedekindDomain.HeightOneSpectrum R} {x : ℚ} :

(IsDedekindDomain.HeightOneSpectrum.valuation ℚ 𝔭) x ≤ 1 ↔ ↑x.den ∉ 𝔭.asIdeal