Theorems · Theorem · commutative algebra

Module.projective_of_localization_maximal

∀ {R : Type u_1} {M : Type uM} [inst : CommRing R] [inst_1 : AddCommGroup M] [inst_2 : Module R M],
  (∀ (I : Ideal R) (x : I.IsMaximal), Module.Projective (Localization.AtPrime I) (LocalizedModule I.primeCompl M)) →
    ∀ [Module.FinitePresentation R M], Module.Projective R M

Defined in: Mathlib.RingTheory.LocalProperties.Projective
Cited by: 3 results in Mathlib
Foundations: Depth 98 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CommRing AddCommGroup Module Module.FinitePresentation

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Cites35

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

DFunLike.coeproof · cited by 62,936
Setproof · cited by 53,352
Modulestatement and proof · cited by 20,661
RingHom.idproof · cited by 18,349
CommRingstatement and proof · cited by 17,173
Finsetproof · cited by 13,712
AddCommGroupstatement and proof · cited by 12,871
LinearMapproof · cited by 10,215
Top.topproof · cited by 9,680
SetLike.coeproof · cited by 8,199
Submoduleproof · cited by 7,192
Finsuppproof · cited by 5,255

Cited by3

Results whose statement or proof uses this declaration.

ModuleCat.hasProjectiveDimensionLE_iff_forall_maximalSpectrumproof · cited by 2
Module.freeLocus_eq_univ_iffproof · cited by 1
Module.projective_of_localization_maximal'proof · cited by 0