Tensor product of algebras

Template:Short descriptionIn 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

<math>A \otimes_R B</math>

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 Template:Nowrap by<ref>Kassel (1995), [[[:Template:Google books]] p. 32].</ref>Template:Sfn

<math>(a_1\otimes b_1)(a_2\otimes b_2) = a_1 a_2\otimes b_1b_2</math>

and then extending by linearity to all of Template:Nowrap. This ring is an R-algebra, associative and unital with the identity element given by Template:Nowrap.<ref>Kassel (1995), [[[:Template:Google books]] p. 32].</ref> 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.Template:Citation needed

Further properties

There are natural homomorphisms from A and B to Template:Nowrap given by<ref>Kassel (1995), [[[:Template:Google books]] p. 32].</ref>

<math>a\mapsto a\otimes 1_B</math>

<math>b\mapsto 1_A\otimes b</math>

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:

<math>\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,</math>

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

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:

<math>X\times_Y Z = \operatorname{Spec}(A\otimes_R B).</math>

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

See also

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

References

• Template:Citation.
• Template:Cite book