Theorems · Theorem · commutative algebra

IsBaseChange.linearMapLeftRightHom_toMatrix

∀ {R : Type u_1} [inst : CommSemiring R] (S : Type u_2) [inst_1 : CommSemiring S] [inst_2 : Algebra R S] (M : Type u_3)
  [inst_3 : AddCommMonoid M] [inst_4 : Module R M] {N : Type u_4} [inst_5 : AddCommMonoid N] [inst_6 : Module R N]
  {P : Type u_5} [inst_7 : AddCommMonoid P] [inst_8 : Module R P] {Q : Type u_6} [inst_9 : AddCommMonoid Q]
  [inst_10 : Module R Q] [inst_11 : Module S P] [inst_12 : IsScalarTower R S P] [inst_13 : Module S Q]
  [inst_14 : IsScalarTower R S Q] {α : M →ₗ[R] P} {β : N →ₗ[R] Q} (ibcM : IsBaseChange S α) (ibcN : IsBaseChange S β)
  {ι : Type u_7} {θ : Type u_8} [inst_15 : DecidableEq ι] [inst_16 : Fintype ι] [inst_17 : Finite θ]
  (b : Module.Basis ι R M) (c : Module.Basis θ R N) (f : M →ₗ[R] N),
  (LinearMap.toMatrix (IsBaseChange.basis b ibcM) (IsBaseChange.basis c ibcN)) ((ibcM.linearMapLeftRightHom β) f) =
    ((LinearMap.toMatrix b c) f).map ⇑(algebraMap R S)

Defined in

Mathlib.RingTheory.TensorProduct.IsBaseChangeHom

Cited by

0 results in Mathlib

Foundations

Depth 103 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringCommSemiringAlgebraAddCommMonoidModuleAddCommMonoidModuleAddCommMonoidModuleAddCommMonoidModuleModuleIsScalarTowerModuleIsScalarTowerDecidableEqFintypeFinite