# Gorenstein properties of simple gluing algebras

## TL;DR

This paper investigates the Gorenstein properties of simple gluing algebras formed from two finite-dimensional bound quiver algebras, establishing conditions for Gorensteinness and describing related categories.

## Contribution

It proves that the Gorenstein property of the glued algebra is equivalent to that of the original algebras and characterizes various categories associated with the glued algebra.

## Key findings

- $ ext{Lambda}$ is Gorenstein iff $A$ and $B$ are Gorenstein.
- Descriptions of Gorenstein projective modules and singularity categories for $ ext{Lambda}$.
- Relations between the Cohen-Macaulay Auslander algebra of $ ext{Lambda}$ and those of $A$ and $B$.

## Abstract

Let $A=KQ_A/I_A$ and $B=KQ_B/I_B$ be two finite-dimensional bound quiver algebras, fix two vertices $a\in Q_A$ and $b\in Q_B$. We define an algebra $\Lambda=KQ_\Lambda/I_\Lambda$, which is called a simple gluing algebra of $A$ and $B$, where $Q_\Lambda$ is from $Q_A$ and $Q_B$ by identifying $a$ and $b$, $I_\Lambda=\langle I_A,I_B\rangle$. We prove that $\Lambda$ is Gorenstein if and only if $A$ and $B$ are Gorenstein, and describe the Gorenstein projective modules, singularity category, Gorenstein defect category and also Cohen-Macaulay Auslander algebra of $\Lambda$ from the corresponding ones of $A$ and $B$.

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Source: https://tomesphere.com/paper/1701.09074