Theorems · Theorem · commutative algebra

Module.associatedPrimes.preimage_comap_associatedPrimes_eq_associatedPrimes_of_isLocalizedModule

∀ {R : Type u_1} [inst : CommRing R] (S : Submonoid R) (R' : Type u_2) [inst_1 : CommRing R'] [inst_2 : Algebra R R']
  [hSR' : IsLocalization S R'] {M : Type u_3} {M' : Type u_4} [inst_3 : AddCommGroup M] [inst_4 : Module R M]
  [inst_5 : AddCommGroup M'] [inst_6 : Module R M'] (f : M →ₗ[R] M') [IsLocalizedModule S f] [inst_8 : Module R' M']
  [IsScalarTower R R' M'] [IsNoetherianRing R],
  Ideal.comap (algebraMap R R') ⁻¹' associatedPrimes R M = associatedPrimes R' M'

[Stacks Tag 0310](https://stacks.math.columbia.edu/tag/0310) ((3))

Defined in

Mathlib.RingTheory.Ideal.AssociatedPrime.Localization

Cited by

1 results in Mathlib

Foundations

Depth 87 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingAlgebraIsLocalizationAddCommGroupModuleAddCommGroupModuleIsLocalizedModuleModuleIsScalarTowerIsNoetherianRing