Theorems · Theorem · commutative algebra
IsLocalization.AtPrime.isDiscreteValuationRing_of_dedekind_domain
∀ (A : Type u_1) [inst : CommRing A] [IsDomain A] [IsDedekindDomain A] {P : Ideal A},
P ≠ ⊥ →
∀ [pP : P.IsPrime] (Aₘ : Type u_2) [inst_3 : CommRing Aₘ] [inst_4 : IsDomain Aₘ] [inst_5 : Algebra A Aₘ]
[IsLocalization.AtPrime Aₘ P], IsDiscreteValuationRing AₘIn a Dedekind domain, the localization at every nonzero prime ideal is a DVR.
- Defined in
- Mathlib.RingTheory.DedekindDomain.Dvr
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 134 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites27
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- Idealstatement and proof · cited by 4,748
- Bot.botstatement and proof · cited by 4,720
- IsDomainstatement and proof · cited by 2,196
- Module.finrankproof · cited by 1,770
- Ideal.IsPrimestatement and proof · cited by 827
- IsDedekindDomainstatement and proof · cited by 668
- Ideal.primeComplproof · cited by 462
- IsLocalRingproof · cited by 339
- IsLocalRing.maximalIdealproof · cited by 297
- ExistsUniqueproof · cited by 268
Cited by2
Results whose statement or proof uses this declaration.
- Ideal.ramificationIdx'_eq_ramificationIdx'proof · cited by 3
- IsLocalization.OverPrime.mem_normalizedFactors_of_isPrimeproof · cited by 1