Theorems · Theorem · commutative algebra

IsLocalizedModule.is_universal

∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} {M' : Type u_3} {M'' : Type u_4}
  [inst_1 : AddCommMonoid M] [inst_2 : AddCommMonoid M'] [inst_3 : AddCommMonoid M''] [inst_4 : Module R M]
  [inst_5 : Module R M'] [inst_6 : Module R M''] (f : M →ₗ[R] M') [IsLocalizedModule S f] (g : M →ₗ[R] M''),
  (∀ (x : ↥S), IsUnit ((algebraMap R (Module.End R M'')) ↑x)) → ∃! l, l ∘ₗ f = g

Universal property from localized module: If (M', f : M ⟶ M') is a localized module then it satisfies the following universal property: For every R-module M'' which every s : S-scalar multiplication is invertible and for every R-linear map g : M ⟶ M'', there is a unique R-linear map l : M' ⟶ M'' such that l ∘ f = g. `` M --f -> M' | / |g / | / l v / M'' ``

Defined in

Mathlib.Algebra.Module.LocalizedModule.Basic

Cited by

3 results in Mathlib

Foundations

Depth 54 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringAddCommMonoidAddCommMonoidAddCommMonoidModuleModuleModuleIsLocalizedModule