Filtrations of Totally Reflexive Modules

TL;DR

This paper introduces a new subcategory of totally reflexive modules characterized by saturated filtrations and upper-triangular presentation matrices, and explores their Ext^1 ranks over a specific ring.

Contribution

It defines a subcategory of totally reflexive modules with saturated filtrations and characterizes them via upper-triangular presentation matrices, advancing understanding of their structure.

Findings

Modules with saturated filtrations have upper-triangular presentation matrices.

Characterization of these modules provides new insights into their structure.

Investigation of Ext^1 ranks over a specific ring reveals their homological properties.

Abstract

In this paper, we will introduce a subcategory of totally reflexive modules that have a saturated filtration by other totally reflexive modules. We will prove these are precisely the totally reflexive modules with an upper-triangular presentation matrix. We conclude with an investigation of the ranks of $\operatorname{Ext}^1$ of two such modules over a specific ring.