Theorems · Definition · linear algebra

homTensorHomEquiv

(R : Type u_2) →
  (M : Type u_3) →
    (N : Type u_4) →
      (P : Type u_5) →
        (Q : Type u_6) →
          [inst : CommSemiring R] →
            [inst_1 : AddCommMonoid M] →
              [inst_2 : AddCommMonoid N] →
                [inst_3 : AddCommMonoid P] →
                  [inst_4 : AddCommMonoid Q] →
                    [inst_5 : Module R M] →
                      [inst_6 : Module R N] →
                        [inst_7 : Module R P] →
                          [inst_8 : Module R Q] →
                            [Module.Projective R M] →
                              [Module.Finite R M] →
                                [Module.Projective R N] →
                                  [Module.Finite R N] →
                                    TensorProduct R (M →ₗ[R] P) (N →ₗ[R] Q) ≃ₗ[R]
                                      TensorProduct R M N →ₗ[R] TensorProduct R P Q

When M and N are free R modules, the map homTensorHomMap is an equivalence. Note that homTensorHomEquiv is not defined directly in terms of homTensorHomMap, but the equivalence between the two is given by homTensorHomEquiv_toLinearMap and homTensorHomEquiv_apply.

Defined in

Mathlib.LinearAlgebra.Contraction

Cited by

2 results in Mathlib

Foundations

Depth 106 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringAddCommMonoidAddCommMonoidAddCommMonoidAddCommMonoidModuleModuleModuleModuleModule.ProjectiveModule.FiniteModule.ProjectiveModule.Finite