Theorems · Definition · linear algebra

TensorProduct.equivOfCompatibleSMul

(R : Type u_1) →
  [inst : CommSemiring R] →
    (A : Type u_22) →
      (S : Type u_23) →
        (M : Type u_24) →
          (N : Type u_25) →
            [inst_1 : AddCommMonoid M] →
              [inst_2 : AddCommMonoid N] →
                [inst_3 : Module R M] →
                  [inst_4 : Module R N] →
                    [inst_5 : CommSemiring A] →
                      [inst_6 : Module A M] →
                        [inst_7 : Module A N] →
                          [SMulCommClass R A M] →
                            [inst_9 : CommSemiring S] →
                              [inst_10 : Module S M] →
                                [inst_11 : SMulCommClass R S M] →
                                  [inst_12 : SMulCommClass A S M] →
                                    [TensorProduct.CompatibleSMul R A M N] →
                                      [TensorProduct.CompatibleSMul A R M N] →
                                        TensorProduct A M N ≃ₗ[S] TensorProduct R M N

If the R- and A-actions on M and N satisfy CompatibleSMul both ways, then M ⊗[A] N is canonically isomorphic to M ⊗[R] N as S-modules, where S is any other ring acting on M and whose action commutes with the A and R-actions.

Defined in

Mathlib.LinearAlgebra.TensorProduct.Basic

Cited by

1 results in Mathlib

Foundations

Depth 63 from the axioms · uses propext, Quot.sound

Assumes

CommSemiringAddCommMonoidAddCommMonoidModuleModuleCommSemiringModuleModuleSMulCommClassCommSemiringModuleSMulCommClassSMulCommClassTensorProduct.CompatibleSMulTensorProduct.CompatibleSMul