# Local homology, finiteness of Tor modules and cofiniteness

## TL;DR

This paper establishes a practical criterion for $a$-cofiniteness of modules over noetherian rings using local homology and finitely generated Tor modules, enhancing computability in algebraic geometry and commutative algebra.

## Contribution

It introduces a new finitely-many-steps criterion for $a$-cofiniteness based on local homology and Tor modules, leveraging local homology theory.

## Key findings

- $a$-cofiniteness characterized by finitely generated Tor modules
- Provides a computable criterion for $a$-cofiniteness
- Highlights the importance of local homology in module theory

## Abstract

Let $\frak a$ be an ideal of a commutative noetherian ring $R$ with unity and $M$ an $R$-module supported at $\V(\fa)$. Let $n$ be the supermum of the integers $i$ for which $H^{\fa}_i(M)\neq 0$. We show that $M$ is $\fa$-cofinite if and only if the $R$-module $\Tor^R_i(R/\fa,M)$ is finitely generated for every $0\leq i\leq n$. This provides a hands-on and computable finitely-many-steps criterion to examine $\mathfrak{a}$-confiniteness. Our approach relies heavily on the theory of local homology which demonstrates the effectiveness and indispensability of this tool.