[Merged by Bors] - feat(RingTheory/TensorProduct): the universal property of the tensor product of algebras #7409
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Rather than showing agreement on all the scalars, it suffices to show agreement on `1`.
…product of algebras
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You're right, we did already have the commutative case (twice!). I've redefined it in terms of the non-commutative case. |
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| variable [CommSemiring R] [CommSemiring S] [Algebra R S] | ||
| variable [Semiring A] [Algebra R A] [Algebra S A] [IsScalarTower R S A] | ||
| variable [Semiring B] [Algebra R B] | ||
| variable [CommSemiring C] [Algebra R C] [Algebra S C] [IsScalarTower R S C] | ||
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| /-- If `A`, `B`, `C` are `R`-algebras, `A` and `C` are also `S`-algebras (forming a tower as | ||
| `·/S/R`), then the product map of `f : A →ₐ[S] C` and `g : B →ₐ[R] C` is an `S`-algebra | ||
| homomorphism. | ||
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| This is just a special case of `Algebra.TensorProduct.lift` for when `C` is commutative. -/ | ||
| abbrev productLeftAlgHom (f : A →ₐ[S] C) (g : B →ₐ[R] C) : A ⊗[R] B →ₐ[S] C := | ||
| lift f g (fun _ _ => Commute.all _ _) | ||
| #align algebra.tensor_product.product_left_alg_hom Algebra.TensorProduct.productLeftAlgHom | ||
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| end |
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Thanks! bors merge |
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…product of algebras (#7409) The construction used for `CliffordAlgebra.ofBaseChange` generalizes into a universal property for building arbitrary algebra morphisms out of the tensor product. It turns out this is a known result; it is mentioned at https://en.wikipedia.org/wiki/Tensor_product_of_algebras#Further_properties which cites the linked reference, though the version added by this PR is heterobasic like the rest of this file. This is a generalization of the existing `Algebra.TensorProduct.productLeftAlgHom` to non-commutative rings.
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The construction used for
CliffordAlgebra.ofBaseChangegeneralizes into a universal property for building arbitrary algebra morphisms out of the tensor product.It turns out this is a known result; it is mentioned at https://en.wikipedia.org/wiki/Tensor_product_of_algebras#Further_properties which cites the linked reference, though the version added by this PR is heterobasic like the rest of this file.
This is a generalization of the existing
Algebra.TensorProduct.productLeftAlgHomto non-commutative rings.