This volume contains expository lectures by Melvin Hochster from the CBMS Regional Conference in Mathematics held at the University of Nebraska, June 1974. The lectures deal mainly with recent developments and still open questions in the homological theory of modules over commutative (usually, Noetherian) rings. A good deal of attention is given to the role 'big' Cohen-Macaulay modules play in clearing up some of the open questions. A modest knowledge of commutative rings and familarity with (the long exact sequences for) Tor and Ext should suffice as a background for the reader.
Introduction Preliminaries Some open questions Consequences of the existence of Cohen-Macaulay modules Modifications and the existence of big Cohen-Macaulay modules in characteristic $p> 0$ Henselian rings, M. Artin's approximation theorem, and big Cohen-Macaulay modules over fields of characteristic 0 Depth-sensitivity theorems and exactness criteria Modules of projective dimension 2, generic modules of finite projective dimension, and the Buchsbaum-Eisenbud structure theorems Linear algebraic groups Applications of homological methods and some more open questions Supplement References.