Theorems · Theorem · commutative algebra

IsLocalizedModule.exists_isLocalizedModule_powers_of_finitePresentation

∀ {R : Type u_3} {M : Type u_4} [inst : CommRing R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] (S : Submonoid R)
  {M' : Type u_1} [inst_3 : AddCommGroup M'] [inst_4 : Module R M'] (f : M →ₗ[R] M') [IsLocalizedModule S f]
  [Module.Finite R M] [Module.FinitePresentation R M'], ∃ r ∈ S, IsLocalizedModule.Away r f

If M is a finite R-module, and the localization Mₛ at some submonoid S of R is finitely presented, then Mₛ = M[1/r] for some r ∈ S.

Defined in

Mathlib.Algebra.Module.FinitePresentation

Cited by

1 results in Mathlib

Foundations

Depth 95 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModuleAddCommGroupModuleIsLocalizedModuleModule.FiniteModule.FinitePresentation