Tensor product of algebras

Further properties

There are natural homomorphisms from A and B to A ⊗_R B given by

${\displaystyle a\mapsto a\otimes 1_B}$

${\displaystyle b\mapsto 1_A\otimes b}$

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:

${\displaystyle \text{Hom}(A\otimes B,X)\cong \{(f,g)\in \text{Hom}(A,X)\times \text{Hom}(B,X)\mid \mathrm{\forall }a\in A,b\in B:[f(a),g(b)]=0\},}$

where [-, -] denotes the commutator. The natural isomorphism is given by identifying a morphism ${\displaystyle \varphi :A\otimes B\to X}$ on the left hand side with the pair of morphisms ${\displaystyle (f,g)}$ on the right hand side where ${\displaystyle f(a):=\varphi (a\otimes 1)}$ and similarly ${\displaystyle g(b):=\varphi (1\otimes b)}$.

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:

${\displaystyle X{\times }_{Y}Z=\mathrm{Spec}(A{\otimes }_{R}B).}$

More generally, the fiber product of schemes is defined by gluing together affine fiber products of this form.

Examples

• The tensor product can be used as a means of taking intersections of two subschemes in a scheme: consider the ${\displaystyle \mathbb{C}[x,y]}$-algebras ${\displaystyle \mathbb{C}[x,y]/f}$, ${\displaystyle \mathbb{C}[x,y]/g}$, then their tensor product is ${\displaystyle \mathbb{C}[x,y]/(f){\otimes }_{\mathbb{C}[x,y]}\mathbb{C}[x,y]/(g)\cong \mathbb{C}[x,y]/(f,g)}$, which describes the intersection of the algebraic curves f = 0 and g = 0 in the affine plane over C.
• More generally, if ${\displaystyle A}$ is a commutative ring and ${\displaystyle I,J\subseteq A}$ are ideals, then ${\displaystyle \frac{A}{I}{\otimes }_A\frac{A}{J}\cong \frac{A}{I+J}}$, with a unique isomorphism sending ${\displaystyle (a+I)\otimes (b+J)}$ to ${\displaystyle ab+I+J}$.
• Tensor products can be used as a means of changing coefficients. For example, ${\displaystyle \mathbb{Z}[x,y]/(x^3+5x^2+x-1){\otimes }_{\mathbb{Z}}\mathbb{Z}/5\cong \mathbb{Z}/5[x,y]/(x^3+x-1)}$ and ${\displaystyle \mathbb{Z}[x,y]/(f){\otimes }_{\mathbb{Z}}\mathbb{C}\cong \mathbb{C}[x,y]/(f)}$.
• Tensor products also can be used for taking products of affine schemes over a field. For example, ${\displaystyle \mathbb{C}[x_1,x_2]/(f(x)){\otimes }_{\mathbb{C}}\mathbb{C}[y_1,y_2]/(g(y))}$ is isomorphic to the algebra ${\displaystyle \mathbb{C}[x_1,x_2,y_1,y_2]/(f(x),g(y))}$ which corresponds to an affine surface in ${\displaystyle {\mathbb{A}}_{\mathbb{C}}^4}$ if f and g are not zero.
• Given ${\displaystyle R}$-algebras ${\displaystyle A}$ and ${\displaystyle B}$ whose underlying rings are graded-commutative rings, the tensor product ${\displaystyle A{\otimes }_{R}B}$ becomes a graded commutative ring by defining ${\displaystyle (a\otimes b)(a^{\prime }\otimes b^{\prime })=(-1{)}^{|b||a^{\prime }|}a{a}^{\prime }\otimes b{b}^{\prime }}$ for homogeneous ${\displaystyle a}$, ${\displaystyle a^{\prime }}$, ${\displaystyle b}$, and ${\displaystyle b^{\prime }}$.
• The tensor product of two matrix algebras is ${\displaystyle M_m(A){\otimes }_A{M}_n(A)\cong M_{mn}(A)}$, the isomorphism given by extending the Kronecker product of two matrices via the universal property.