MODULES WITH DESCENDING CHAIN CONDITION

duality for complete, local rings developed there. The results of the present paper guarantee the existence of a sufficient quantity of modules with D.C.C. and provide a basis for a link between the theory of such modules and the theory of finitely generated ones. The Koszul complex, with its dual nature, plays an important role in establishing this link. In ?1 we introduce the functors X and XM. By considering these functors we are able to give characterizations of modules with maximal orders; and decompose them uniquely into direct sums, where each summand depends on only a single maximal ideal. We then prove a transition theorem which enables us to pass to rings of quotients and their completions. A key result of this section is the theorem that if A is an R-module with D.C.C., and if I is an ideal of R, then IA =A if and only if there exists an element reI such that rA =A. This is the dual of a standard result for finitely generated modules. In ?2 we introduce the concepts of cosequences and dimension, primarily for modules with D.C.C. These concepts are naturally dual to the concepts of sequences and codimension for finitely generated modules. The cohomology of the Koszul complex is exploited to provide results concerning cosequences that are dual to those obtained for sequences in [2] from the homology of the Koszul complex. Furthermore, a relationship is determined between cosequences and injective dimension that is dual to the relationship between sequences and projective dimension found in [1]. The two theories are actually equivalent, yield the same global information about the ring R, and either theory may be used to determine the other. In fact, the dependence of the codimension of R on its maximal ideals finds its strongest expression in the above mentioned duality. In ?3 we examine the projective dimension of modules with maximal orders. We also generalize a theorem of D. Rees [5], and obtain a direct