Theorems · Theorem · commutative algebra

IsLocalization.AtPrime.isDedekindDomain

∀ (A : Type u_1) [inst : CommRing A] [IsDomain A] [IsDedekindDomain A] (P : Ideal A) [inst_3 : P.IsPrime]
  (Aₘ : Type u_2) [inst_4 : CommRing Aₘ] [IsDomain Aₘ] [inst_6 : Algebra A Aₘ] [IsLocalization.AtPrime Aₘ P],
  IsDedekindDomain Aₘ

The localization of a Dedekind domain at every nonzero prime ideal is a Dedekind domain.

Defined in: Mathlib.RingTheory.DedekindDomain.Dvr
Cited by: 1 results in Mathlib
Foundations: Depth 120 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CommRing IsDomain IsDedekindDomain Ideal.IsPrime CommRing IsDomain Algebra IsLocalization.AtPrime

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Cites9

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
IsDomainstatement and proof · cited by 2,196
Ideal.IsPrimestatement and proof · cited by 827
IsDedekindDomainstatement and proof · cited by 668
IsLocalization.AtPrimestatement and proof · cited by 79
Ideal.primeCompl_le_nonZeroDivisorsproof · cited by 17
IsLocalization.isDedekindDomainproof · cited by 1

Cited by1

Results whose statement or proof uses this declaration.

IsLocalization.AtPrime.isDiscreteValuationRing_of_dedekind_domainproof · cited by 2