Bass and Betti numbers of a module and its deficiency modules

TL;DR

This paper explores the relationships between Bass and Betti numbers of modules and their deficiency modules, providing generalizations of existing results, characterizations of Cohen-Macaulay modules, and insights into the Auslander-Reiten conjecture.

Contribution

It introduces new relations between Bass and Betti numbers and applies these to generalize known results and analyze Cohen-Macaulay modules and the Auslander-Reiten conjecture.

Findings

Established relations between Bass and Betti numbers of modules and deficiency modules.

Generalized results of Foxby regarding Cohen-Macaulay modules.

Provided a case for the Auslander-Reiten conjecture.

Abstract

This paper aims to provide several relations between Bass and Betti numbers of a given module and its deficiency modules. Such relations and the tools used throughout allow us to generalize some results of Foxby, characterize Cohen-Macaulay modules in equidimensionality terms and furnish a case for the Auslander-Reiten conjecture.