TORSION THEORY, CO-COHEN-MACAULAY AND LOCAL HOMOLOGY

Let A be a commutative ring and M an Artinian .A-module. Let $\sigma$ be a torsion radical functor and (T, F) it's corresponding partition of Spec(A) In [1] the concept of Cohen-Macauly modules was generalized . In this paper we shall define $\sigma$-co-Cohen-Macaulay (abbr. $\sigma$-co-CM). Indeed this is one of the aims of this paper, we obtain some satisfactory properties of such modules. An-other aim of this paper is to generalize the concept of cograde by using the left derived functor $U^{\alpha}$$_{I}$(-) of the $\alpha$-adic completion functor, where a is contained in Jacobson radical of A.A.