Gorenstein properties of simple gluing algebras

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 $B$.