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
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
[1]
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 1A ⊗ 1B.
[3] where 1A and 1B 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]
Further properties
There are natural homomorphisms from A and B to A ⊗R B given by
[4]
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:
where [-, -] denotes the
commutator.
The
natural isomorphism is given by identifying a morphism on the left hand side with the pair of morphisms on the right hand side where and similarly .
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:
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 -algebras , , then their tensor product is , 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 are ideals, then , with a unique isomorphism sending to .
- Tensor products can be used as a means of changing coefficients. For example, and .
- Tensor products also can be used for taking
products of affine schemes over a field. For example, is
isomorphic to the algebra which corresponds to an affine surface in if f and g are not zero.
- Given -algebras and whose underlying rings are
graded-commutative rings, the tensor product becomes a graded commutative ring by defining for homogeneous , , , and .
See also
Notes
References