In mathematics, the tensor product of two algebras over a commutative ring R is also an R-algebra. This gives the tensor product of algebras. When the ring is a field, the most common application of such products is to describe the product of algebra representations.
Definition Let R be a commutative ring and let A and B be R-algebras. Since A and B may both be regarded as R-modules, their tensor product
A ⊗ R B {\displaystyle A\otimes _{R}B}
is also an R-module. The tensor product can be given the structure of a ring by defining the product on elements of the form a ⊗ b by
( a 1 ⊗ b 1 ) ( a 2 ⊗ b 2 ) = a 1 a 2 ⊗ b 1 b 2 {\displaystyle (a_{1}\otimes b_{1})(a_{2}\otimes b_{2})=a_{1}a_{2}\otimes b_{1}b_{2}}
and then extending by linearity to all of A ⊗R B. This ring is an R-algebra, associative and unital with the identity element given by 1A ⊗ 1B, where 1A and 1B are the identity elements of A and B. If A and B are commutative, then the tensor product is commutative as well. The tensor product turns the category of R-algebras into a symmetric monoidal category.
Further properties There are natural homomorphisms from A and B to A ⊗R B given by
a ↦ a ⊗ 1 B {\displaystyle a\mapsto a\otimes 1_{B}}
b ↦ 1 A ⊗ b {\displaystyle b\mapsto 1_{A}\otimes b}
These maps make the tensor product the coproduct in the category of commutative R-algebras. The tensor product is not the coproduct in the category of all R-algebras; there the coproduct is given by a more general free product of algebras. Nevertheless, the tensor product of non-commutative algebras can be described by a universal property similar to that of the coproduct:
Hom ( A ⊗ B , X ) ≅ { ( f , g ) ∈ Hom ( A , X ) × Hom ( B , X ) ∣ ∀ a ∈ A , b ∈ B : [ f ( a ) , g ( b ) ] = 0 } , {\displaystyle {\text{Hom}}(A\otimes B,X)\cong \lbrace (f,g)\in {\text{Hom}}(A,X)\times {\text{Hom}}(B,X)\mid \forall a\in A,b\in B:[f(a),g(b)]=0\rbrace ,}
where [-, -] denotes the commutator. The natural isomorphism is given by identifying a morphism ϕ : A ⊗ B → X {\displaystyle \phi :A\otimes B\to X} on the left hand side with the pair of morphisms ( f , g ) {\displaystyle (f,g)} on the right hand side where f ( a ) := ϕ ( a ⊗ 1 ) {\displaystyle f(a):=\phi (a\otimes 1)} and similarly g ( b ) := ϕ ( 1 ⊗ b ) {\displaystyle g(b):=\phi (1\otimes b)} .
Applications The tensor product of commutative algebras is of frequent use in algebraic geometry. For affine schemes X, Y, Z with morphisms from X and Z to Y, so X = Spec(A), Y = Spec(R), and Z = Spec(B) for some commutative rings A, R, B, the fiber product scheme is the affine scheme corresponding to the tensor product of algebras:
X × Y Z = Spec ( A ⊗ R B ) . {\displaystyle X\times _{Y}Z=\operatorname {Spec} (A\otimes _{R}B).}
More generally, the fiber product of schemes is defined by gluing together affine fiber products of this form.
Examples
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