Torsion theory, co-Cohen-Macaulay and local homology

Abstract

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. Another aim of this paper is to generalize the concept of cograde by using the left derived functor $U_i^{\fa}(-)$ of the $\fa$-adic completion functor, where $\fa$ is contained in Jacobson radical of $A$.