The notion of a Cohen–Macaulay ring marks the cross–roads of two powerful lines of research in present-day commutative algebra. While its main development belongs to the homological theory of commutative rings, it finds surprising and fruitful applications in the realm of algebraic combinatorics. Consequently this book is an introduction to the homological and combinatorial aspects of commutative algebra.

We have tried to keep the text self-contained. However, it has not proved possible, and would perhaps not have been appropriate, to develop commutative ring theory from scratch. Instead we assume the reader has acquired some fluency in the language of rings, ideals, and modules by working through an introductory text like Atiyah and Macdonald [15] or Sharp [344]. Nevertheless, to ease the access for the non-expert, the essentials of dimension theory have been collected in an appendix.

As exemplified by Matsumura's standard textbook [270], it is natural to have the notions of grade and depth follow dimension theory, and so Chapter 1 opens with the introduction of regular sequences on which their definition is based. From the very beginning we stress their connection with homological and linear algebra, and in particular with the Koszul complex.

Chapter 2 introduces Cohen–Macaulay rings and modules, our main subjects. Next we study regular local rings. They form the most special class of Cohen–Macaulay rings; their theory culminates in the Auslander–Buchsbaum–Serre and Auslander–Buchsbaum–Nagata theorems. Unlike the Cohen–Macaulay property in general, regularity has a very clear geometric interpretation: it is the algebraic counterpart of the notion of a non–singular point. Similarly the third class of rings introduced in Chapter 2, that of complete intersections, is of geometric significance.