Discrete valuation rings

This file defines discrete valuation rings (DVRs) and develops a basic interface for them.

Important definitions

There are various definitions of a DVR in the literature; we define a DVR to be a local PID which is not a field (the first definition in Wikipedia) and prove that this is equivalent to being a PID with a unique non-zero prime ideal (the definition in Serre's book "Local Fields").

Let R be an integral domain, assumed to be a principal ideal ring and a local ring.

• DiscreteValuationRing R : a predicate expressing that R is a DVR.

Definitions

• addVal R : AddValuation R PartENat : the additive valuation on a DVR.

Implementation notes

It's a theorem that an element of a DVR is a uniformizer if and only if it's irreducible. We do not hence define Uniformizer at all, because we can use Irreducible instead.

Tags

discrete valuation ring

class DiscreteValuationRing (R : Type u) [CommRing R] [IsDomain R] extends IsPrincipalIdealRing , LocalRing : Prop

An integral domain is a discrete valuation ring (DVR) if it's a local PID which is not a field.

• principal : ∀ (S : Ideal R), Submodule.IsPrincipal S
• exists_pair_ne : ∃ (x : R) (y : R), x ≠ y
• isUnit_or_isUnit_of_add_one : ∀ {a b : R}, a + b = 1 → IsUnit a ∨ IsUnit b
• not_a_field' : LocalRing.maximalIdeal R ≠ ⊥

theorem DiscreteValuationRing.not_a_field (R : Type u) [CommRing R] [IsDomain R] [DiscreteValuationRing R] : LocalRing.maximalIdeal R ≠ ⊥

theorem DiscreteValuationRing.not_isField (R : Type u) [CommRing R] [IsDomain R] [DiscreteValuationRing R] : ¬IsField R

A discrete valuation ring R is not a field.

theorem DiscreteValuationRing.irreducible_of_span_eq_maximalIdeal {R : Type u_1} [CommRing R] [LocalRing R] [IsDomain R] (ϖ : R) (hϖ : ϖ ≠ 0) (h : LocalRing.maximalIdeal R = Ideal.span {ϖ}) : Irreducible ϖ

theorem DiscreteValuationRing.irreducible_iff_uniformizer {R : Type u} [CommRing R] [IsDomain R] [DiscreteValuationRing R] (ϖ : R) : Irreducible ϖ ↔ LocalRing.maximalIdeal R = Ideal.span {ϖ}

An element of a DVR is irreducible iff it is a uniformizer, that is, generates the maximal ideal of R.

theorem Irreducible.maximalIdeal_eq {R : Type u} [CommRing R] [IsDomain R] [DiscreteValuationRing R] {ϖ : R} (h : Irreducible ϖ) : LocalRing.maximalIdeal R = Ideal.span {ϖ}

theorem DiscreteValuationRing.exists_irreducible (R : Type u) [CommRing R] [IsDomain R] [DiscreteValuationRing R] : ∃ (ϖ : R), Irreducible ϖ

Uniformizers exist in a DVR.

theorem DiscreteValuationRing.exists_prime (R : Type u) [CommRing R] [IsDomain R] [DiscreteValuationRing R] : ∃ (ϖ : R), Prime ϖ

Uniformizers exist in a DVR.

theorem DiscreteValuationRing.iff_pid_with_one_nonzero_prime (R : Type u) [CommRing R] [IsDomain R] : DiscreteValuationRing R ↔ IsPrincipalIdealRing R ∧ ∃! P : Ideal R, P ≠ ⊥ ∧ Ideal.IsPrime P

An integral domain is a DVR iff it's a PID with a unique non-zero prime ideal.

theorem DiscreteValuationRing.associated_of_irreducible (R : Type u) [CommRing R] [IsDomain R] [DiscreteValuationRing R] {a : R} {b : R} (ha : Irreducible a) (hb : Irreducible b) : Associated a b

def DiscreteValuationRing.HasUnitMulPowIrreducibleFactorization (R : Type u_1) [CommRing R] : Prop

Alternative characterisation of discrete valuation rings.

Equations

• DiscreteValuationRing.HasUnitMulPowIrreducibleFactorization R = ∃ (p : R), Irreducible p ∧ ∀ {x : R}, x ≠ 0 → ∃ (n : ℕ), Associated (p ^ n) x

theorem DiscreteValuationRing.HasUnitMulPowIrreducibleFactorization.unique_irreducible {R : Type u_1} [CommRing R] (hR : DiscreteValuationRing.HasUnitMulPowIrreducibleFactorization R) ⦃p : R⦄ ⦃q : R⦄ (hp : Irreducible p) (hq : Irreducible q) : Associated p q

theorem DiscreteValuationRing.HasUnitMulPowIrreducibleFactorization.toUniqueFactorizationMonoid {R : Type u_1} [CommRing R] (hR : DiscreteValuationRing.HasUnitMulPowIrreducibleFactorization R) [IsDomain R] : UniqueFactorizationMonoid R

An integral domain in which there is an irreducible element p such that every nonzero element is associated to a power of p is a unique factorization domain. See DiscreteValuationRing.ofHasUnitMulPowIrreducibleFactorization.

theorem DiscreteValuationRing.HasUnitMulPowIrreducibleFactorization.of_ufd_of_unique_irreducible {R : Type u_1} [CommRing R] [IsDomain R] [UniqueFactorizationMonoid R] (h₁ : ∃ (p : R), Irreducible p) (h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q) : DiscreteValuationRing.HasUnitMulPowIrreducibleFactorization R

theorem DiscreteValuationRing.aux_pid_of_ufd_of_unique_irreducible (R : Type u) [CommRing R] [IsDomain R] [UniqueFactorizationMonoid R] (h₁ : ∃ (p : R), Irreducible p) (h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q) : IsPrincipalIdealRing R

theorem DiscreteValuationRing.of_ufd_of_unique_irreducible {R : Type u} [CommRing R] [IsDomain R] [UniqueFactorizationMonoid R] (h₁ : ∃ (p : R), Irreducible p) (h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q) : DiscreteValuationRing R

A unique factorization domain with at least one irreducible element in which all irreducible elements are associated is a discrete valuation ring.

theorem DiscreteValuationRing.ofHasUnitMulPowIrreducibleFactorization {R : Type u} [CommRing R] [IsDomain R] (hR : DiscreteValuationRing.HasUnitMulPowIrreducibleFactorization R) : DiscreteValuationRing R

An integral domain in which there is an irreducible element p such that every nonzero element is associated to a power of p is a discrete valuation ring.

theorem DiscreteValuationRing.associated_pow_irreducible {R : Type u_1} [CommRing R] [IsDomain R] [DiscreteValuationRing R] {x : R} (hx : x ≠ 0) {ϖ : R} (hirr : Irreducible ϖ) : ∃ (n : ℕ), Associated x (ϖ ^ n)

theorem DiscreteValuationRing.eq_unit_mul_pow_irreducible {R : Type u_1} [CommRing R] [IsDomain R] [DiscreteValuationRing R] {x : R} (hx : x ≠ 0) {ϖ : R} (hirr : Irreducible ϖ) : ∃ (n : ℕ) (u : Rˣ), x = ↑u * ϖ ^ n

theorem DiscreteValuationRing.ideal_eq_span_pow_irreducible {R : Type u_1} [CommRing R] [IsDomain R] [DiscreteValuationRing R] {s : Ideal R} (hs : s ≠ ⊥) {ϖ : R} (hirr : Irreducible ϖ) : ∃ (n : ℕ), s = Ideal.span {ϖ ^ n}

theorem DiscreteValuationRing.unit_mul_pow_congr_pow {R : Type u_1} [CommRing R] [IsDomain R] [DiscreteValuationRing R] {p : R} {q : R} (hp : Irreducible p) (hq : Irreducible q) (u : Rˣ) (v : Rˣ) (m : ℕ) (n : ℕ) (h : ↑u * p ^ m = ↑v * q ^ n) : m = n

theorem DiscreteValuationRing.unit_mul_pow_congr_unit {R : Type u_1} [CommRing R] [IsDomain R] [DiscreteValuationRing R] {ϖ : R} (hirr : Irreducible ϖ) (u : Rˣ) (v : Rˣ) (m : ℕ) (n : ℕ) (h : ↑u * ϖ ^ m = ↑v * ϖ ^ n) : u = v

The additive valuation on a DVR

noncomputable def DiscreteValuationRing.addVal (R : Type u) [CommRing R] [IsDomain R] [DiscreteValuationRing R] : AddValuation R PartENat

The PartENat-valued additive valuation on a DVR.

Equations

• DiscreteValuationRing.addVal R = multiplicity.addValuation ⋯

theorem DiscreteValuationRing.addVal_def {R : Type u_1} [CommRing R] [IsDomain R] [DiscreteValuationRing R] (r : R) (u : Rˣ) {ϖ : R} (hϖ : Irreducible ϖ) (n : ℕ) (hr : r = ↑u * ϖ ^ n) : (DiscreteValuationRing.addVal R) r = ↑n

theorem DiscreteValuationRing.addVal_def' {R : Type u_1} [CommRing R] [IsDomain R] [DiscreteValuationRing R] (u : Rˣ) {ϖ : R} (hϖ : Irreducible ϖ) (n : ℕ) : (DiscreteValuationRing.addVal R) (↑u * ϖ ^ n) = ↑n

theorem DiscreteValuationRing.addVal_zero {R : Type u_1} [CommRing R] [IsDomain R] [DiscreteValuationRing R] : (DiscreteValuationRing.addVal R) 0 = ⊤

theorem DiscreteValuationRing.addVal_one {R : Type u_1} [CommRing R] [IsDomain R] [DiscreteValuationRing R] : (DiscreteValuationRing.addVal R) 1 = 0

@[simp]

theorem DiscreteValuationRing.addVal_uniformizer {R : Type u_1} [CommRing R] [IsDomain R] [DiscreteValuationRing R] {ϖ : R} (hϖ : Irreducible ϖ) : (DiscreteValuationRing.addVal R) ϖ = 1

theorem DiscreteValuationRing.addVal_mul {R : Type u_1} [CommRing R] [IsDomain R] [DiscreteValuationRing R] {a : R} {b : R} : (DiscreteValuationRing.addVal R) (a * b) = (DiscreteValuationRing.addVal R) a + (DiscreteValuationRing.addVal R) b

theorem DiscreteValuationRing.addVal_pow {R : Type u_1} [CommRing R] [IsDomain R] [DiscreteValuationRing R] (a : R) (n : ℕ) : (DiscreteValuationRing.addVal R) (a ^ n) = n • (DiscreteValuationRing.addVal R) a

theorem Irreducible.addVal_pow {R : Type u_1} [CommRing R] [IsDomain R] [DiscreteValuationRing R] {ϖ : R} (h : Irreducible ϖ) (n : ℕ) : (DiscreteValuationRing.addVal R) (ϖ ^ n) = ↑n

theorem DiscreteValuationRing.addVal_eq_top_iff {R : Type u_1} [CommRing R] [IsDomain R] [DiscreteValuationRing R] {a : R} : (DiscreteValuationRing.addVal R) a = ⊤ ↔ a = 0

theorem DiscreteValuationRing.addVal_le_iff_dvd {R : Type u_1} [CommRing R] [IsDomain R] [DiscreteValuationRing R] {a : R} {b : R} : (DiscreteValuationRing.addVal R) a ≤ (DiscreteValuationRing.addVal R) b ↔ a ∣ b

theorem DiscreteValuationRing.addVal_add {R : Type u_1} [CommRing R] [IsDomain R] [DiscreteValuationRing R] {a : R} {b : R} : min ((DiscreteValuationRing.addVal R) a) ((DiscreteValuationRing.addVal R) b) ≤ (DiscreteValuationRing.addVal R) (a + b)

instance DiscreteValuationRing.instIsHausdorffMaximalIdealToCommSemiringToLocalRingToAddCommGroupToRingToModuleToSemiring (R : Type u_2) [CommRing R] [IsDomain R] [DiscreteValuationRing R] : IsHausdorff (LocalRing.maximalIdeal R) R

Equations

• ⋯ = ⋯