The notion of depth as defined in Chapter 0 is a valuable concept in commutative algebra. In case of modules of finite projective dimension, one has an invariant on which one may apply mathematical induction. However, if the ring in question is not regular, there are finitely generated modules of infinite projective dimension. In 1957 Auslander and Buchsbaum pointed out that depth, which they called “codimension”, is complimentary to projective dimension for finitely generated modules of finite projective dimension. Moreover, they established that finitely generated modules over any local ring have finite depth. Thus depth gives an invariant for which, with the aid of induction, one can hope to obtain results about all finitely generated modules over a local ring. The landmark papers of Peskine and Szpiro (1973) and of Hochster (1976) exploited the notion of depth and its relation to dimension in order to solve a host of problems which had evolved from attempts to tackle questions from classical ideal theory using the techniques of homological algebra. Typical of these results is the intersection theorem (1.13) which carries the classical intersection theorem for smooth varieties over to the nonsmooth case.

In order to obtain results for finitely generated modules we follow Hochster's lead and broaden our scope to include infinitely generated modules. Thus we will need a concept of depth that works for infinitely generated modules and that will reduce to the definition given in Chapter 0 for the case of finitely generated modules.