In this chapter we will review some basic facts without proofs and give some of the basic notation that will be used throughout the book. For further results in commutative algebra we refer the reader to the excellent textbooks of Matsumura, and Nagata. For material such as local cohomologies and canonical modules, we recommend Herzog and Kunz.

Throughout this chapter R is a commutative Noetherian local ring with maximal ideal m and with residue field k = R/m. We always denote the Krull dimension of R by d. All modules considered here will be finitely generated and unitary.

CM modules

Let M be an R-module. Recall that a sequence {x1, x2 … xn} of elements in m is a regular sequence on M if xi+1 is a non zero divisor on M(x1, x2, … xi, …) M for any i (0 ≤ i, < n). The depth of M is the maximum length of regular sequences on M.

In this book we shall be concerned exclusively with Cohen-Macaulay modules, which are defined as follows:

(1.1) DEFINITION. An R-module M is called a maximal Cohen-Macaulay module or simply a Cohen-Macaulay (abbr. CM) module if the depth of M is equal to d. The ring R is a CM ring if R is a CM module over R.

The reader may recall several equivalent definitions of CM modules.