Theorems · Theorem · commutative algebra

Ideal.spanIntNorm_localization

∀ (R : Type u_1) [inst : CommRing R] [inst_1 : IsDomain R] {S : Type u_3} [inst_2 : CommRing S] [inst_3 : IsDomain S]
  [inst_4 : IsIntegrallyClosed R] [inst_5 : IsIntegrallyClosed S] [inst_6 : Algebra R S] [inst_7 : Module.Finite R S]
  [inst_8 : Module.IsTorsionFree R S] (I : Ideal S),
  ∀ M ≤ nonZeroDivisors R,
    ∀ {Rₘ : Type u_4} (Sₘ : Type u_5) [inst_9 : CommRing Rₘ] [inst_10 : Algebra R Rₘ] [inst_11 : CommRing Sₘ]
      [inst_12 : Algebra S Sₘ] [inst_13 : Algebra Rₘ Sₘ] [inst_14 : Algebra R Sₘ] [IsScalarTower R Rₘ Sₘ]
      [IsScalarTower R S Sₘ] [IsLocalization M Rₘ] [IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ]
      [inst_19 : IsIntegrallyClosed Rₘ] [inst_20 : IsDomain Rₘ] [inst_21 : IsDomain Sₘ]
      [inst_22 : Module.IsTorsionFree Rₘ Sₘ] [inst_23 : Module.Finite Rₘ Sₘ] [inst_24 : IsIntegrallyClosed Sₘ],
      Ideal.spanNorm Rₘ (Ideal.map (algebraMap S Sₘ) I) = Ideal.map (algebraMap R Rₘ) (Ideal.spanNorm R I)

Defined in

Mathlib.RingTheory.Ideal.Norm.RelNorm

Cited by

3 results in Mathlib

Foundations

Depth 188 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingIsDomainCommRingIsDomainIsIntegrallyClosedIsIntegrallyClosedAlgebraModule.FiniteModule.IsTorsionFreeCommRingAlgebraCommRingAlgebraAlgebraAlgebraIsScalarTowerIsScalarTowerIsLocalizationIsLocalizationIsIntegrallyClosedIsDomainIsDomainModule.IsTorsionFreeModule.FiniteIsIntegrallyClosed