Theorems · Theorem · commutative algebra

exists_bijective_map_powers

∀ {R : Type u_3} {M : Type u_4} {N : Type u_5} [inst : CommRing R] [inst_1 : AddCommGroup M] [inst_2 : Module R M]
  [inst_3 : AddCommGroup N] [inst_4 : Module R N] (S : Submonoid R) {M' : Type u_1} [inst_5 : AddCommGroup M']
  [inst_6 : Module R M'] (f : M →ₗ[R] M') [inst_7 : IsLocalizedModule S f] {N' : Type u_2} [inst_8 : AddCommGroup N']
  [inst_9 : Module R N'] (g : N →ₗ[R] N') [inst_10 : IsLocalizedModule S g] [Module.Finite R M]
  [Module.FinitePresentation R N] (l : M →ₗ[R] N),
  Function.Bijective ⇑((IsLocalizedModule.map S f g) l) →
    ∃ r ∈ S, ∀ (t : R), r ∣ t → Function.Bijective ⇑((LocalizedModule.map (Submonoid.powers t)) l)

Let M be a finite R-module, and N be a finitely presented R-module. If l : M →ₗ[R] N is a linear map whose localization at S : Submonoid R is bijective, then l is already bijective under the localization at some r ∈ S.

Defined in

Mathlib.Algebra.Module.FinitePresentation

Cited by

3 results in Mathlib

Foundations

Depth 94 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModuleAddCommGroupModuleAddCommGroupModuleIsLocalizedModuleAddCommGroupModuleIsLocalizedModuleModule.FiniteModule.FinitePresentation