Theorems · Theorem · commutative algebra

IsLocalizedModule.map_linearCombination

∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) (A : Type u_2) [inst_1 : CommSemiring A]
  [inst_2 : Algebra R A] [inst_3 : 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') {α : Type u_7} {v : α → M} [inst_10 : IsLocalizedModule S f],
  (IsLocalizedModule.map S (Finsupp.mapRange.linearMap (Algebra.linearMap R A)) f) (Finsupp.linearCombination R v) =
    ↑R (Finsupp.linearCombination A (⇑f ∘ v))

Defined in

Mathlib.RingTheory.Localization.BaseChange

Cited by

3 results in Mathlib

Foundations

Depth 98 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringCommSemiringAlgebraIsLocalizationAddCommMonoidModuleAddCommMonoidModuleModuleIsScalarTowerIsLocalizedModule