Theorems · Definition · ring theory

Algebra.TensorProduct.algEquivOfLinearEquivTensorProduct

{R : Type uR} →
  {S : Type uS} →
    {A : Type uA} →
      {B : Type uB} →
        {C : Type uC} →
          [inst : CommSemiring R] →
            [inst_1 : CommSemiring S] →
              [inst_2 : Algebra R S] →
                [inst_3 : Semiring A] →
                  [inst_4 : Algebra R A] →
                    [inst_5 : Algebra S A] →
                      [inst_6 : IsScalarTower R S A] →
                        [inst_7 : Semiring B] →
                          [inst_8 : Algebra R B] →
                            [inst_9 : Semiring C] →
                              [inst_10 : Algebra S C] →
                                (f : TensorProduct R A B ≃ₗ[S] C) →
                                  (∀ (a₁ a₂ : A) (b₁ b₂ : B),
                                      f ((a₁ * a₂) ⊗ₜ[R] (b₁ * b₂)) = f (a₁ ⊗ₜ[R] b₁) * f (a₂ ⊗ₜ[R] b₂)) →
                                    f (1 ⊗ₜ[R] 1) = 1 → TensorProduct R A B ≃ₐ[S] C

Build an algebra equivalence from a linear equivalence out of a tensor product, and evidence that on pure tensors, it preserves multiplication and the identity. Note that we state h_one using 1 ⊗ₜ[R] 1 instead of 1 so that lemmas about f applied to pure tensors can be directly applied by the caller (without needing TensorProduct.one_def).

Defined in

Mathlib.RingTheory.TensorProduct.Maps

Cited by

1 results in Mathlib

Foundations

Depth 79 from the axioms · uses propext, Quot.sound

Assumes

CommSemiringCommSemiringAlgebraSemiringAlgebraAlgebraIsScalarTowerSemiringAlgebraSemiringAlgebra