Theorems · Theorem · commutative algebra

Module.injective_of_localization_maximal

∀ {R : Type u} [inst : CommRing R] {M : Type v} [inst_1 : AddCommGroup M] [inst_2 : Module R M] [Small.{v, u} R]
  [IsNoetherianRing R],
  (∀ (I : Ideal R) (x : I.IsMaximal), Module.Injective (Localization.AtPrime I) (LocalizedModule I.primeCompl M)) →
    Module.Injective R M

Defined in: Mathlib.RingTheory.LocalProperties.Injective
Cited by: 2 results in Mathlib
Foundations: Depth 96 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CommRing AddCommGroup Module Small IsNoetherianRing

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Cites53

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

DFunLike.coeproof · cited by 62,936
Modulestatement and proof · cited by 20,661
RingHom.idproof · cited by 18,349
CommRingstatement and proof · cited by 17,173
AddCommGroupstatement and proof · cited by 12,871
LinearMapproof · cited by 10,215
Idealstatement and proof · cited by 4,748
Algebra.algebraMapproof · cited by 4,706
LinearMap.compproof · cited by 1,642
LinearEquiv.symmproof · cited by 1,461
LinearEquiv.toLinearMapproof · cited by 1,171
LinearMap.extproof · cited by 844

Cited by2

Results whose statement or proof uses this declaration.

ModuleCat.hasInjectiveDimensionLE_iff_forall_maximalSpectrumproof · cited by 2
Module.injective_of_localization_maximal'proof · cited by 0