Tensor product of algebras

{{Short description|Tensor product of algebras over a field; itself another algebra}}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 {{nowrap|a ⊗ b}} byKassel (1995), [{{Google books|plainurl=y|id=S1KE_pToY98C|page=32|text=we put an algebra structure on the tensor product}} p. 32].{{sfn|Lang|2002|pp=629-630}}

: $(a_1\otimes b_1)(a_2\otimes b_2) = a_1 a_2\otimes b_1b_2$

and then extending by linearity to all of {{nowrap|A ⊗_R B}}. This ring is an R-algebra, associative and unital with the identity element given by {{nowrap|1_A ⊗ 1_B}}.Kassel (1995), [{{Google books|plainurl=y|id=S1KE_pToY98C|page=32|text=Its unit is}} p. 32]. 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.{{citation needed|date=October 2015}}

Further properties

There are natural homomorphisms from A and B to {{nowrap|A ⊗_R B}} given byKassel (1995), [{{Google books|plainurl=y|id=S1KE_pToY98C|page=32|text=get algebra morphisms}} p. 32].

: $a\mapsto a\otimes 1_B$

: $b\mapsto 1_A\otimes b$

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:

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

where [-, -] denotes the commutator.

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

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:

: $X\times_Y Z = \operatorname{Spec}(A\otimes_R B).$

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

Examples

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, $\mathbb{Z}[x,y]/(x^3 + 5x^2 + x - 1)\otimes_\mathbb{Z} \mathbb{Z}/5 \cong \mathbb{Z}/5[x,y]/(x^3 + x - 1)$ and $\mathbb{Z}[x,y]/(f) \otimes_\mathbb{Z} \mathbb{C} \cong \mathbb{C}[x,y]/(f)$ .
Tensor products also can be used for taking products of affine schemes over a field. For example, $\mathbb{C}[x_1,x_2]/(f(x)) \otimes_\mathbb{C} \mathbb{C}[y_1,y_2]/(g(y))$ 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 $\mathbb{A}^4_\mathbb{C}$ if f and g are not zero.
Given $R$ -algebras $A$ and $B$ whose underlying rings are graded-commutative rings, the tensor product $A\otimes_RB$ becomes a graded commutative ring by defining $(a\otimes b)(a'\otimes b')=(-1)^$
b a'
aa'\otimes bb' for homogeneous $a$ , $a'$ , $b$ , and $b'$ .

Notes

References

{{Citation| last1=Kassel| first1=Christian| date=1995| title=Quantum groups| volume=155| series=Graduate texts in mathematics| publisher=Springer| isbn=978-0-387-94370-1| url-access=registration| url=https://archive.org/details/quantumgroups0000kass}}.
{{cite book |last=Lang |first=Serge |title=Algebra |series=Graduate Texts in Mathematics |volume=21 |publisher=Springer |year=2002 |orig-year=first published in 1993 |ISBN=0-387-95385-X }}

Category:Algebras

Category:Ring theory

Category:Commutative algebra

Category:Multilinear algebra