Local Homology, Koszul Homology and Serre Classes
Kamran Divaani-Aazar, Hossein Faridian, Massoud Tousi

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 of modules, we compare the containment of the Koszul homology, Ext modules, Tor modules, local homology, and local cohomology in up to a given bound . 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.
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Taxonomy
TopicsCommutative Algebra and Its Applications · Algebraic structures and combinatorial models · Homotopy and Cohomology in Algebraic Topology
