On the finiteness of local homology modules

Abstract

Let $R$ be a commutative Noetherian ring and $\mathfrak{a}$ be an ideal of $R$. Suppose $M$ is a finitely generated $R$-module and $N$ is an Artinian $R$-module. We define the concept of filter coregular sequence to determine the infimum of integers $i$ such that the generalized local homology ${\text{H}}_i^{\mathfrak{a}}(M,N)$ is not finitely generated as an ${\hat{R}}^{\mathfrak{a}}$-module, where ${\hat{R}}^{\mathfrak{a}}$ denotes the $\mathfrak{a}$-adic completion of $R$. In particular, if $R$ is a complete semi-local ring, then ${\text{H}}_i^{\mathfrak{a}}(M,N)$ is a finitely generated ${\hat{R}}^{\mathfrak{a}}$-module for all non-negative integers $i$ if and only if $(0{:}_N\mathfrak{a}+\text{Ann}(M))$ has finite length.

arXiv commentThis paper is accepted for publication in the Journal of Algebraic Systems