Multilinear Algebra

Abstract

Polynomial rings form a simple example of a graded algebra; such algebras occur frequently and in Section 6.1 we define this concept . Another important example is given by free algebras, which are discussed in Section 6.2, as well as the related notions of tensor algebra and symmetric algebra on a K-module . A graded algebra has an important invariant, its Hilbert series, essentially a power series whose coefficients indicate the dimensions of the components. In Section 6.3 we show that the Hilbert series of a commutative Noetherian ring is a rational function and also prove the Golod-Shafarevich theorem, giving a sufficient condition for a graded algebra to be infinite-dimensional. The applications, to construct a finitely generated algebra which is nil but not nilpotent, and a finitely generated infinite p-group , are sketched in the exercises. Finally Section 6.4 deals with exterior algebras, providing a simple derivation of determinants, and giving a brief geometrical application.