Theorems · Definition · linear algebra

lTensorHomEquivHomLTensor

(R : Type u_2) →
  (M : Type u_3) →
    (P : Type u_5) →
      (Q : Type u_6) →
        [inst : CommSemiring R] →
          [inst_1 : AddCommMonoid M] →
            [inst_2 : AddCommMonoid P] →
              [inst_3 : AddCommMonoid Q] →
                [inst_4 : Module R M] →
                  [inst_5 : Module R P] →
                    [inst_6 : Module R Q] →
                      [Module.Projective R M] →
                        [Module.Finite R M] → TensorProduct R P (M →ₗ[R] Q) ≃ₗ[R] M →ₗ[R] TensorProduct R P Q

When M is a finite free module, the map lTensorHomToHomLTensor is an equivalence. Note that lTensorHomEquivHomLTensor is not defined directly in terms of lTensorHomToHomLTensor, but the equivalence between the two is given by lTensorHomEquivHomLTensor_toLinearMap and lTensorHomEquivHomLTensor_apply.

Defined in

Mathlib.LinearAlgebra.Contraction

Cited by

4 results in Mathlib

Foundations

Depth 105 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringAddCommMonoidAddCommMonoidAddCommMonoidModuleModuleModuleModule.ProjectiveModule.Finite