Gorenstein properties of simple gluing algebras

Abstract

Let $A=K{Q}_A/I_A$ and $B=K{Q}_B/I_B$ be two finite-dimensional bound quiver algebras, fix two vertices $a\in Q_A$ and $b\in Q_B$. We define an algebra $\Lambda =K{Q}_{\Lambda }/I_{\Lambda }$, 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 $b$, $I_{\Lambda }=⟨I_A,I_B⟩$. 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 $B$.

22 pages, 5 figures