Review of Commutative Algebra

Abstract

In this chapter we recall basis concepts from commutative algebra which are relevant for the subjects treated in the later chapters. We begin with a review on graded rings, Hilbert functions, and Hilbert series, and introduce the multiplicity and the a-invariant of a graded module. The Krull dimension of a graded module will be defined in terms of its Hilbert series. We will give various characterizations of the depth of a module and its relation to the Krull dimension. These considerations lead to Cohen–Macaulay modules and Gorenstein rings. We then describe the relationship, known as Auslander–Buchsbaum formula, between the depth of a graded S-module M and its projective dimension, where S is a polynomial ring, and study in more detail the finite minimal graded free S-resolution of M. The regularity of M will be defined via this resolution. Koszul algebras are standard graded K-algebras whose graded maximal ideal has a linear resolution. Unless this graded ring is a polynomial ring, this resolution is infinite. We discuss various necessary and sufficient conditions for Koszulness. The methods involved include Grobner bases and Koszul filtrations.