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.

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 $B$.