Theorems · Definition · ring theory

Algebra.TensorProduct.mapOfCompatibleSMul

(R : Type u_3) →
  (S : Type u_4) →
    (T : Type u_5) →
      (A : Type u_6) →
        (B : Type u_7) →
          [inst : CommSemiring R] →
            [inst_1 : CommSemiring S] →
              [inst_2 : CommSemiring T] →
                [inst_3 : Semiring A] →
                  [inst_4 : Semiring B] →
                    [inst_5 : Algebra R A] →
                      [inst_6 : Algebra R B] →
                        [inst_7 : Algebra S A] →
                          [inst_8 : Algebra S B] →
                            [inst_9 : Algebra T A] →
                              [inst_10 : SMulCommClass R T A] →
                                [inst_11 : SMulCommClass S T A] →
                                  [SMulCommClass R S A] →
                                    [TensorProduct.CompatibleSMul R S A B] →
                                      TensorProduct S A B →ₐ[T] TensorProduct R A B

If A and B are both R- and S-algebras and their actions on them commute, and if the S-action on A ⊗[R] B can switch between the two factors, then there is a canonical T-algebra homomorphism from A ⊗[S] B to A ⊗[R] B, where T is any other ring acting on A and whose action commutes with the R and S-actions.

Defined in

Mathlib.RingTheory.TensorProduct.Maps

Cited by

8 results in Mathlib

Foundations

Depth 78 from the axioms · uses propext, Quot.sound

Assumes

CommSemiringCommSemiringCommSemiringSemiringSemiringAlgebraAlgebraAlgebraAlgebraAlgebraSMulCommClassSMulCommClassSMulCommClassTensorProduct.CompatibleSMul