Theorems · Theorem · commutative algebra

IsDedekindDomain.HeightOneSpectrum.valuation_div_le_one_iff

∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsDedekindDomain R] (K : Type u_2) [inst_2 : Field K]
  [inst_3 : Algebra R K] [inst_4 : IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) (a : R) {b : R},
  b ≠ 0 →
    (b ∈ v.asIdeal → a ∉ v.asIdeal) → ((IsDedekindDomain.HeightOneSpectrum.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.

Defined in

Mathlib.RingTheory.DedekindDomain.AdicValuation

Cited by

2 results in Mathlib

Foundations

Depth 160 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingIsDedekindDomainFieldAlgebraIsFractionRing