On the extension-closed property for the subcategory $\text{Tr}(\Omega^2({\rm mod}-A))$

TL;DR

This paper constructs a specific monomial quiver algebra to demonstrate that the subcategory formed by second syzygies of modules is not always extension-closed, providing a counterexample to a previously open question.

Contribution

It provides the first known example of a monomial quiver algebra where the subcategory of second syzygies is not extension-closed, answering a question by Reiten.

Findings

Counterexample to extension-closed property for $ ext{Tr}( ext{Omega}^2( ext{mod}-A))$

Shows that extension-closedness does not hold universally for this subcategory

Advances understanding of homological properties in monomial quiver algebras

Abstract

We present a monomial quiver algebra $A$ having the property that the subcategory $\text{Tr}(\Omega^2({\rm mod}-A))$ is not extension-closed. This answers a question raised by Idun Reiten.