Tensor product of algebras explained

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 ⊗ _RB

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 by^[1]

(a_{1 ⊗}b_1)(a_{2 ⊗}b₂₎=a₁a_{2 ⊗}b_1b₂

and then extending by linearity to all of . This ring is an R-algebra, associative and unital with identity element given by .^[2] 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 given by^[3]

a\mapstoa ⊗ 1_B

b\mapsto1_{A ⊗}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)\cong\lbrace(f,g)\inHom(A,X) x Hom(B,X)\mid\foralla\inA,b\inB:[f(a),g(b)]=0\rbrace,

where [-, -] denotes the commutator.The natural isomorphism is given by identifying a morphism

\phi:A ⊗ B\toX

on the left hand side with the pair of morphisms

(f,g)

on the right hand side where

f(a):=\phi(a ⊗ 1)

and similarly

g(b):=\phi(1 ⊗ 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 x _YZ=\operatorname{Spec}(A ⊗ _RB).

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

C[x,y]

-algebras

C[x,y]/f

C[x,y]/g

, then their tensor product is

C[x,y]/(f) ⊗ _C[x,y]C[x,y]/(g)\congC[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

is a commutative ring and

I,J\subseteqA

are ideals, then

	A
	I

⊗

A	A
	J

\cong

	A
	I+J

, with a unique isomorphism sending

(a+I) ⊗ (b+J)

(ab+I+J)

Tensor products can be used as a means of changing coefficients. For example,

Z[x,y]/(x³+5x²+x-1) ⊗ _ZZ/5\congZ/5[x,y]/(x³+x-1)

and

Z[x,y]/(f) ⊗ _ZC\congC[x,y]/(f)

Tensor products also can be used for taking products of affine schemes over a field. For example,

C[x_1,x_2]/(f(x)) ⊗ _CC[y_1,y_2]/(g(y))

is isomorphic to the algebra

C[x_1,x_2,y_1,y_{2]/(f(x),g(y))}

which corresponds to an affine surface in

	4
A
	C

if f and g are not zero.

Given

-algebras

and

whose underlying rings are graded-commutative rings, the tensor product

A ⊗ _RB

becomes a graded commutative ring by defining

(a ⊗ b)(a' ⊗ b')=(-1)^|b||a'|aa' ⊗ bb'

for homogeneous

, and

References

.
Book: Lang, Serge . Algebra . Graduate Texts in Mathematics . 21 . Springer . 2002 . first published in 1993 . 0-387-95385-X .

Notes and References

Kassel (1995), [{{Google books|plainurl=y|id=S1KE_pToY98C|page=32|text=we put an algebra structure on the tensor product}} p. 32].
Kassel (1995), [{{Google books|plainurl=y|id=S1KE_pToY98C|page=32|text=Its unit is}} p. 32].
Kassel (1995), [{{Google books|plainurl=y|id=S1KE_pToY98C|page=32|text=get algebra morphisms}} p. 32].