Local Homology, Koszul Homology and Serre Classes

TL;DR

This paper explores the relationships between various homology and cohomology modules within Serre classes, providing characterizations, vanishing results, and connections to invariants like depth and width.

Contribution

It offers a comprehensive comparison of homological modules within Serre classes and characterizes local homology modules, unifying several vanishing criteria and invariants.

Findings

Characterization of noetherian local homology modules

Vanishing results linking Koszul, local homology, and cohomology

Unified descriptions of depth and width via homological modules

Abstract

Given a Serre class $\mathcal{S}$ of modules, we compare the containment of the Koszul homology, Ext modules, Tor modules, local homology, and local cohomology in $\mathcal{S}$ up to a given bound $s \geq 0$. As some applications, we give a full characterization of noetherian local homology modules. Further, we establish a comprehensive vanishing result which readily leads to the formerly known descriptions of the numerical invariants width and depth in terms of Koszul homology, local homology, and local cohomology. Also, we immediately recover a few renowned vanishing criteria scattered about the literature.