On generalized completion homology modules
Let $I$ be an ideal of a commutative Noetherian ring $R$. Let $M$ and $N$ be any $R$-modules. We define the generalized completion homology modules $L_i\Lambda^I (N,M)$, for $i\in \mathbb{Z}$, as the homologies of the complex $\lim\limits_{\longleftarrow}(N/I^sN\otimes_R F_{\cdot}^R)$. Here $F_{\cdot}^R$ denote a flat resolution of $M$. In this article we will prove the vanishing and non-vanishing properties of $L_i\Lambda^I (N,M)$. We denote $H^{i}_{I}(N,M)$ (resp. $U^I_i(N,M)$) by the generalized local cohomology modules (resp. the generalized local homology modules). As a technical tool we will construct several natural homomorphisms of $L_i\Lambda^I (N,M)$, $H^{i}_{I}(N,M)$ and $U^I_i(N,M)$. We will investigate when these natural homomorphisms are isomorphisms. Moreover if $M$ is Artinian and $N$ is finitely generated then it is proven that $L_i\Lambda^I (N,M)$ is isomorphic to $U^I_i(N,M)$ for each $i\in \mathbb{Z}$. The similar result is obtained for $H^i_{I}(N,M)$. Furthermore if both $M$ and $N$ are finitely generated with $c=\grade(I,M)$. Then we are able to prove several necessary and sufficient conditions such that $H^i_{I}(M)=0$ for all $i\neq c.$ Here $H^i_{I}(M)$ denote the ordinary local cohomology module.