On the finiteness of local homology modules

TL;DR

This paper investigates the conditions under which generalized local homology modules are finitely generated, introducing filter coregular sequences and linking finiteness to the length of certain submodules in complete semi-local rings.

Contribution

It defines filter coregular sequences to analyze the finiteness of local homology modules and establishes a criterion relating finiteness to module length in complete semi-local rings.

Findings

Generalized local homology modules are finitely generated iff a specific submodule has finite length in complete semi-local rings.

Introduces filter coregular sequences to study the finiteness properties of local homology modules.

Provides a characterization of the non-finiteness of local homology modules in terms of module length.

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 $\textrm{H}^{\mathfrak{a}}_i(M, N)$ is not finitely generated as an $\widehat{R}^{\mathfrak{a}}$-module, where $\widehat{R}^{\mathfrak{a}}$ denotes the $\mathfrak{a}$-adic completion of $R$. In particular, if $R$ is a complete semi-local ring, then $\textrm{H}^{\mathfrak{a}}_i(M, N)$ is a finitely generated $\widehat{R}^{\mathfrak{a}}$-module for all non-negative integers $i$ if and only if $(0:_N\mathfrak{a}+\textrm{Ann}(M))$ has finite length.