Since its birth inD. Hilbert’s paramount article ”Über die Theorie der algebraischen Formen” [Math. Ann. 36, 473–534 (1890;JFM 22.0133.01)] syzygy theory has always been a main theme of commutative algebra. Within syzygy theory, the author’s ”hearts and theorems lie predominantly with syzygies of finite projective dimension over local rings”.
After a review of some basic facts in chapter 0 the authors prepare for the homological investigation of syzygies in chapter 1 by studying the notion of depth and the pertaining consequences of the existence of maximal Cohen-Macaulay modules. Chapter 2 details the construction of basic elements and the limitations thereof given by ”principal ideal theorems”. The heart of the book is chapter 3, covering the fundamental results and culminating in the author’s syzygy theorem. Chapter 4 contains applications like the factoriality of regular local rings, the characterization of rings with small multiplicities, and the existence of rings with prescribed local cohomology. Chapter 5 is devoted to a generalization of Serre’s trick of killing Ext\(^ 1\). In the last chapter the authors relate syzygy theory and problems concerning vector bundles on the punctured spectrum of a regular local ring, the main result covered being their theorem that a bundle with at least one vanishing cohomology module cannot be lifted indefinitely. The appendix describes the construction of some well-known vector bundles of small rank.
Many important results on the structure of ideals and modules are collected along the road towards the main theorems, partly in the text itself, partly in the numerous exercises accompanying each chapter. The book is written in a very lively style. Presupposing only a modest knowledge of commutative algebra, the authors guide the reader swiftly into one of its central areas.

Reviewer: W.Bruns