# An application of a theorem of Sheila Brenner for Hochschild extension algebras of a truncated quiver algebra

## TL;DR

This paper applies Sheila Brenner's theorem to analyze Hochschild extension algebras of truncated quiver algebras, determining the structure of almost split sequences in this context.

## Contribution

It provides a formula for the number of indecomposable summands in the middle term of almost split sequences for specific Hochschild extension algebras.

## Key findings

- Number of indecomposable summands explicitly determined
- Extension algebras analyzed via Brenner's theorem
- Applicable to algebras with zero cycles in the quiver

## Abstract

Let $A$ be a truncated quiver algebra over an algebraically closed field such that any oriented cycle in the ordinary quiver of $A$ is zero in $A$. We give the number of the indecomposable direct summands of the middle term of an almost split sequence for a class of Hochschild extension algebras of $A$ by the standard duality module $D(A)$.