### Co-Cohen-Macaulay Artinian modules over commutative rings

I. H. Denizler, R. Y. Sharp
1996 Glasgow Mathematical Journal
0. Introduction. In [7] , Z. Tang and H. Zakeri introduced the concept of co-Cohen-Macaulay Artinian module over a quasi-local commutative ring R (with identity): a non-zero Artinian R-module A is said to be a co-Cohen-Macaulay module if and only if codepth A = dim ,4, where codepth ^4 is the length of a maximal /4-cosequence and dim/4 is the Krull dimension of A as defined by R. N. Roberts in [2] . Tang and Zakeri obtained several properties of co-Cohen-Macaulay Artinian ^-modules, including a
more » ... characterization of such modules by means of the modules of generalized fractions introduced by Zakeri and the present second author in [6] ; this characterization is explained as follows. Let m denote the maximal ideal of R, and let A be a non-zero Artinian /?-module of Krull dimension d > 0. Roberts [2, Theorem 6] proved that d is equal to the least integer /' for which there exists a proper ideal q of R generated by i elements such that (O^q) has finite length. A sequence a"... ,a d of d elements of m is called a system of parameters (s.o.p.) for A if (0: A (a u ... ,a d )) has finite length. For an integer; with l^j^d and b u ... ,bj e R, we say that the sequence b\,...,bj is a partial system of parameters A. Tang and Zakeri set, for each i e M (we use N (respectively N o ) to denote the set of positive (respectively non-negative) integers), U t : = {(r u ..., r t ) e R'-.there exists ; with 0 =£; =s i such that r"... , r, is a p.s.o.p. for A and r ;+1 = . . . = /-, = 1}. They showed [7, Theorem 3.10] that °U = (U n ) neN is a chain of triangular sets on R in the sense of L. O'Carroll [1, p. 420] (so that the complex C(% R) of modules of generalized fractions can be formed, as described in [1, p. 420]), and that A is co-Cohen-Macaulay if and only if the complex Hom R (C(% R),A) is exact. Tang and Zakeri did not extend the notion of co-Cohen-Macaulay module to Artinian modules over arbitrary commutative rings, and so we make here the following obvious definition. 0.1. DEFINITION. Let A be a non-zero Artinian module over the commutative ring R (with identity). Observe that A m is an Artinian /? m -module for each maximal ideal m E Supp(/l). We say that A is a co-Cohen-Macaulay 7?-module precisely when A m is a co-Cohen-Macaulay fl m -module for each maximal ideal m e Supp(y4). It should be noted that, in the situation of this definition, the support of the Artinian -module A consists of maximal ideals; furthermore, it is an elementary exercise to show that Supp(/4) is equal to the finite set of maximal ideals m of R for which Soc(/4), the socle of A, has a submodule isomorphic to R/m. One of the aims of this paper is to provide a generalization of Tang's and Zakerfs