Tensor product of algebras

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

$A{\otimes }_{R}B$

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

$(a_1\otimes b_1)(a_2\otimes b_2)=a_1{a}_2\otimes b_1{b}_2$

and then extending by linearity to all of A ⊗_R B. This ring is an R-algebra, associative and unital with identity element given by 1_A ⊗ 1_B. where 1_A and 1_B 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

$a\mapsto a\otimes 1_B$

$b\mapsto 1_A\otimes b$

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:

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

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

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:

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

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

Examples

See also: tensor product of modules § Examples

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

See also

• Extension of scalars
• Tensor product of modules
• Tensor product of fields
• Linearly disjoint
• Multilinear subspace learning

References

• Kassel, Christian (1995), Quantum groups, Graduate texts in mathematics, vol. 155, Springer, ISBN 978-0-387-94370-1.
• Lang, Serge (2002) [first published in 1993]. Algebra. Graduate Texts in Mathematics. Vol. 21. Springer. ISBN 0-387-95385-X.