Theorems · Theorem · commutative algebra

IsDedekindDomain.primesOverEquivPrimesOver_symm_apply

∀ {R : Type u_1} {S : Type u_2} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] (p : Ideal R)
  [inst_3 : p.IsPrime] (Rₚ : Type u_3) [inst_4 : CommRing Rₚ] [inst_5 : Algebra R Rₚ]
  [inst_6 : IsLocalization.AtPrime Rₚ p] [inst_7 : IsLocalRing Rₚ] (Sₚ : Type u_4) [inst_8 : CommRing Sₚ]
  [inst_9 : Algebra S Sₚ] [inst_10 : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ]
  [inst_11 : Algebra Rₚ Sₚ] [inst_12 : IsDomain R] [inst_13 : IsDedekindDomain S] [inst_14 : Module.IsTorsionFree R S]
  [inst_15 : Algebra R Sₚ] [inst_16 : IsScalarTower R S Sₚ] [inst_17 : IsScalarTower R Rₚ Sₚ] (hp : p ≠ ⊥)
  (Q : ↑((IsLocalRing.maximalIdeal Rₚ).primesOver Sₚ)),
  ↑((IsDedekindDomain.primesOverEquivPrimesOver p Rₚ Sₚ hp).symm Q) = Ideal.comap (algebraMap S Sₚ) ↑Q

Defined in

Mathlib.RingTheory.Localization.AtPrime.Extension

Cited by

0 results in Mathlib

Foundations

Depth 139 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingAlgebraIdeal.IsPrimeCommRingAlgebraIsLocalization.AtPrimeIsLocalRingCommRingAlgebraIsLocalizationAlgebraIsDomainIsDedekindDomainModule.IsTorsionFreeAlgebraIsScalarTowerIsScalarTower