Documentation

Mathlib.RingTheory.DedekindDomain.AdicValuation.Valuation

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.

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.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 #

Tags #

dedekind domain, dedekind ring, adic valuation

Adic valuations on the Dedekind domain R #

noncomputable 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.

Equations

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

Instances For

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.

noncomputable 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.

Equations

v.intValuation = { toFun := v.intValuationDef, map_zero' := ⋯, map_one' := ⋯, map_mul' := ⋯, map_add_le_max' := ⋯ }

Instances For

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

v.intValuation r = WithZero.exp (-↑(multiplicity v.asIdeal (Ideal.span {r})))

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_eq_one_iff_mem_primeCompl {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : HeightOneSpectrum R) (r : R) :

v.intValuation r = 1 ↔ r ∈ v.asIdeal.primeCompl

The v-adic valuation of r : R is equal to 1 if and only if r ∈ vᶜ.

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

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

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

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

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).

instance IsDedekindDomain.HeightOneSpectrum.instIsNontrivialWithZeroMultiplicativeIntIntValuation {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : HeightOneSpectrum R) :

v.intValuation.IsNontrivial

@[simp]

theorem IsDedekindDomain.HeightOneSpectrum.intValuation_uniformizer {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : HeightOneSpectrum R) (π : v.intValuation.Uniformizer) :

v.intValuation ↑π.val = 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` #

noncomputable 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.

Equations

IsDedekindDomain.HeightOneSpectrum.valuation K v = v.intValuation.extendToLocalization ⋯ K

Instances For

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.

@[simp]

theorem IsDedekindDomain.HeightOneSpectrum.valuation_eq_one_iff_notMem {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) ((algebraMap R K) r) = 1 ↔ r ∉ v.asIdeal

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

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

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

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

Function.Injective (valuation K)

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

Function.Injective fun (p : HeightOneSpectrum R) => (valuation K p).valuationSubring

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

IsDedekindDomain ↥(Localization.subalgebra.ofField K v.asIdeal.primeCompl ⋯)

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

IsLocalRing ↥(Localization.subalgebra.ofField K v.asIdeal.primeCompl ⋯)

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

ValuationSubring K

Given a Dedekind domain R in K, its field of fractions, the localization of R at a nonzero prime is a valuation subring of K.

Equations

IsDedekindDomain.HeightOneSpectrum.valuationSubringAtPrime K v = ValuationSubring.ofSubring (Localization.subalgebra.ofField K v.asIdeal.primeCompl ⋯).toSubring ⋯

Instances For

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

(valuationSubringAtPrime K v).toSubring = (Localization.subalgebra.ofField K v.asIdeal.primeCompl ⋯).toSubring

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

valuationSubringAtPrime K v ≤ (valuation K v).valuationSubring

@[instance_reducible]

noncomputable instance IsDedekindDomain.HeightOneSpectrum.instAlgebraSubtypeMemValuationSubringValuationSubringAtPrime {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 ↥(valuationSubringAtPrime K v)

Equations

v.instAlgebraSubtypeMemValuationSubringValuationSubringAtPrime = (Localization.subalgebra.ofField K v.asIdeal.primeCompl ⋯).algebra'

instance IsDedekindDomain.HeightOneSpectrum.instIsScalarTowerSubtypeMemValuationSubringValuationSubringAtPrime {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 (↥(valuationSubringAtPrime K v)) K

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

IsDedekindDomain ↥(valuationSubringAtPrime K v)

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

Ring.KrullDimLE 1 ↥(valuationSubringAtPrime K v)

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

IsLocalization v.asIdeal.primeCompl ↥(valuationSubringAtPrime K v)

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

valuationSubringAtPrime K v = (valuation K v).valuationSubring

Given v : HeightOneSpectrum R, the valuation associated to v has the localization of R at v as valuation subring.

theorem IsDedekindDomain.HeightOneSpectrum.exists_primeCompl_mul_eq_or_mul_eq {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) :

∃ (n : R) (d : ↥v.asIdeal.primeCompl), x * (algebraMap R K) ↑d = (algebraMap R K) n ∨ x * (algebraMap R K) n = (algebraMap R K) ↑d

All x : K can be written as n / d or d / n with n : R and d ∈ v.asIdealᶜ.

theorem IsDedekindDomain.HeightOneSpectrum.exists_primeCompl_mul_eq_of_integer {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) (hv : (valuation K v) x ≤ 1) :

∃ (n : R) (d : ↥v.asIdeal.primeCompl), x * (algebraMap R K) ↑d = (algebraMap R K) n

All x ∈ 𝓞[K] can be written as n / d with n : R and d ∈ v.asIdealᶜ.

theorem IsDedekindDomain.HeightOneSpectrum.exists_intValuation_mul_sub_lt {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : HeightOneSpectrum R) {a b : R} (hv : v.intValuation b ≤ v.intValuation a) (γ : Multiplicative ℤ) :

∃ (y : R), v.intValuation (b - y * a) < ↑γ

Given a, b ∈ A and v b ≤ v a we can find y : A such that y * a is close to b by the valuation v.

theorem IsDedekindDomain.HeightOneSpectrum.exists_valuation_sub_lt_of_integer {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} (hv : (valuation K v) x ≤ 1) (γ : (WithZero (Multiplicative ℤ))ˣ) :

∃ (a : R), (valuation K v) ((algebraMap R K) a - x) < ↑γ

Given x ∈ 𝒪[K] we can find a : A such that a is close to x by the valuation v.

noncomputable 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

Equations

v.intAdicAbvDef hb r = (WithZeroMulInt.toNNReal ⋯) (v.intValuation r)

Instances For

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)

noncomputable 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

Equations

v.intAdicAbv hb = { toFun := fun (r : R) => ↑(v.intAdicAbvDef hb r), map_mul' := ⋯, nonneg' := ⋯, eq_zero' := ⋯, add_le' := ⋯ }

Instances For

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

noncomputable 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

Equations

v.adicAbvDef hb x = (WithZeroMulInt.toNNReal ⋯) ((IsDedekindDomain.HeightOneSpectrum.valuation K v) x)

Instances For

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)

noncomputable 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

Equations

v.adicAbv hb = { toFun := fun (x : K) => ↑(v.adicAbvDef hb x), map_mul' := ⋯, nonneg' := ⋯, eq_zero' := ⋯, add_le' := ⋯ }

Instances For

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