Appendix A. Tensors, Filtrations, and Gradings

Abstract

If $E$ is a vector space, the tensor algebra $T(E)$ of $E$ is the direct sum over $n\geq0$ of the $n$-fold tensor product of $E$ with itself. This is an associative algebra with a universal mapping property relative to any linear mapping of $E$ into an associative algebra $A$ with identity: the linear map extends to an algebra homomorphism of $T(E)$ into $A$ carrying 1 into 1. Also any linear map of $E$ into $T(E)$ extends to a derivation of $T(E)$.

The symmetric algebra $S(E)$ is a quotient of $T(E)$ with the following universal mapping property: any linear mapping of $E$ into a commutative associative algebra $A$ with identity extends to an algebra homomorphism of $S(E)$ into $A$ carrying 1 into 1. The symmetric algebra is commutative.

Similarly the exterior algebra $\bigwedge(E)$ is a quotient of $T(E)$ with this universal mapping property: any linear mapping $l$ of $E$ into an associative algebra $A$ with identity such that $l(v)^2=0$ for all $v\in E$ extends to an algebra homomorphism of $\bigwedge(E)$ into $A$ carrying 1 into 1.

The tensor algebra, the symmetric algebra, and the exterior algebra are all examples of graded associative algebras. A more general notion than a graded algebra is that of a filtered algebra. A filtered associative algebra has an associated graded algebra. The notions of gradings and filtrations make sense in the context of vector spaces, and a linear map between filtered vector spaces that respects the filtration induces an associated graded map between the associated graded vector spaces. If the associated graded map is an isomorphism, then the original map is an isomorphism.

A ring with identity is left Noetherian if its left ideals satisfy the ascending chain condition. If a filtered algebra is given and if the associated graded algebra is left Noetherian, then the filtered algebra itself is left Noetherian.