On generalized completion homology modules

TL;DR

This paper introduces generalized completion homology modules for modules over Noetherian rings, studies their properties, and explores conditions under which they relate to local cohomology and homology, extending classical results.

Contribution

It defines new generalized completion homology modules, investigates their vanishing, non-vanishing, and isomorphism properties, and relates them to local cohomology and homology in various module contexts.

Findings

Vanishing and non-vanishing properties of generalized completion homology modules established.

Isomorphism between generalized completion homology and local homology for Artinian and finitely generated modules proven.

Conditions for the vanishing of local cohomology modules outside a specific degree identified.

Abstract

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.