Theorems · Theorem · commutative algebra

isLocalizedModule_iff_isBaseChange

∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) (A : Type u_2) [inst_1 : CommSemiring A]
  [inst_2 : Algebra R A] [IsLocalization S A] {M : Type u_3} [inst_4 : AddCommMonoid M] [inst_5 : Module R M]
  {M' : Type u_4} [inst_6 : AddCommMonoid M'] [inst_7 : Module R M'] [inst_8 : Module A M']
  [inst_9 : IsScalarTower R A M'] (f : M →ₗ[R] M'), IsLocalizedModule S f ↔ IsBaseChange A f

The map (f : M →ₗ[R] M') is a localization of modules iff the map (Localization S) × M → N, (s, m) ↦ s • f m is the tensor product (insomuch as it is the universal bilinear map). In particular, there is an isomorphism between LocalizedModule S M and (Localization S) ⊗[R] M given by m/s ↦ (1/s) ⊗ₜ m.

Defined in

Mathlib.RingTheory.Localization.BaseChange

Cited by

6 results in Mathlib

Foundations

Depth 73 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringCommSemiringAlgebraIsLocalizationAddCommMonoidModuleAddCommMonoidModuleModuleIsScalarTower