Suppose that is a commutative ring with unity. Let be an module.
If
then we can define a (non-commutative) product map
by defining
This allows us to view the infinite direct sum
as a (non-commutative) -algebra. This ring is called the tensor algebra of .
Universal property
If is any -algebra and is any module, then given a map there is a unique map which, when restricted to gives .
Graded rings
The tensor algebra is an example of a graded ring. A general graded ring is any ring that is a direct sum of additive subgroups where . Each is called the subset of homogeneous elements of degree .
A graded ideal is an ideal of that is the direct sum of its homogeneous components .
A map between graded rings is a homomorphism of graded rings if for every .
Example: The polynomial ring is graded where the homogeneous components are spanned by the monomials of a given degree. The ideal generated by is graded because any element of this ideal – that is, any polynomiial with zero constant term – can be written as a sum of monomials of a fixed degree in a unique way.
The ring is graded by degree, and the ideal generated by is not graded because and are not in .
An ideal generated by homogeneous elements in a graded ring is graded.
Quotients of graded rings
If is a graded ring and is a graded ideal, then is graded with homogeneous components .
The symmetric algebra
The symmetric algebra - definition
A bilinear map is symmetric if for all pairs . The symmetric tensor product has the universal property that any symmetric bilinar map “factors through” in a unique way. is constructed from by imposing the relation for all pairs .
The symmetric algebra is obtained from the tensor algebra by modding out by the ideal generated by all in . The map
is a map of graded rings since is a graded ideal, so is a graded ring. and .
More on the symmetric algebra
In , the degree terms are spanned by tensors modulo the relation which allows you to freely permute the terms.
Any -symmetric multilinear map factors through in a unique way.
If is any commutative algebra, and is a module map, then there is a unique -algebra map which restricts to on .
If is a field and is a vector space of dimension then is the (graded) polynomial ring in variables over . If is a commutative ring and is a free module of rank then is the (graded) polynomial ring over in variables.
The exterior algebra
Alternating maps and the wedge product
A bilinear map is alternating if for all . The exterior product (or wedge product) has the property that there is a map such that if is an alternating map, then there is a unique map making the usual triangle commute.
The map is written and is spanned by all . We have for elementary tensors where .
is the quotient of by the elementary tensors .
The exterior algebra
The quotient of the tensor algebra by the ideal generated by all tensors of the form is called the exterior algebra, written . It is graded so that that the degree terms consist of linear combinations of the “wedge” of elements of .
If is an -algebra such that for all , and is a module homomorphism, then there is a unique map which restricts to .
If is a finite dimensional vector space over a field , then is finite dimensional. This holds for any free, finite rank module over a ring .
If , if then is spanned by so . If , then there is an alternating map from to defined by
so is not zero.
The cross product in .
The coefficients of
in terms of the two-forms give the cross product in .