Theorems · Theorem · commutative algebra

Module.Basis.ofIsLocalizedModule_span

∀ {R : Type u_1} (Rₛ : Type u_2) [inst : CommSemiring R] (S : Submonoid R) [inst_1 : CommSemiring Rₛ]
  [inst_2 : Algebra R Rₛ] [inst_3 : IsLocalization S Rₛ] {M : Type u_3} {Mₛ : Type u_4} [inst_4 : AddCommMonoid M]
  [inst_5 : Module R M] [inst_6 : AddCommMonoid Mₛ] [inst_7 : Module R Mₛ] [inst_8 : Module Rₛ Mₛ]
  [inst_9 : IsScalarTower R Rₛ Mₛ] (f : M →ₗ[R] Mₛ) [inst_10 : IsLocalizedModule S f] {ι : Type u_5}
  (b : Module.Basis ι R M), Submodule.span R (Set.range ⇑(Module.Basis.ofIsLocalizedModule Rₛ S f b)) = f.range

Defined in

Mathlib.RingTheory.Localization.Module

Cited by

2 results in Mathlib

Foundations

Depth 91 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringCommSemiringAlgebraIsLocalizationAddCommMonoidModuleAddCommMonoidModuleModuleIsScalarTowerIsLocalizedModule