Silting Modules over Triangular Matrix Rings

Abstract

Let $\Lambda$, $\Gamma$ be rings and $R = \left( \begin{smallmatrix} \Lambda & 0 \\ M & \Gamma \end{smallmatrix} \right)$ the triangular matrix ring with $M$ a $(\Gamma,\Lambda)$-bimodule. Let $X$ be a right $\Lambda$-module and $Y$ a right $\Gamma$-module. We prove that $(X,0) \oplus (Y \otimes_{\Gamma} M, Y)$ is a silting right $R$-module if and only if both $X_{\Lambda}$ and $Y_{\Gamma}$ are silting modules and $Y \otimes_{\Gamma} M$ is generated by $X$. Furthermore, we prove that if $\Lambda$ and $\Gamma$ are finite dimensional algebras over an algebraically closed field and $X_{\Lambda}$ and $Y_{\Gamma}$ are finitely generated, then $(X,0) \oplus (Y \otimes_{\Gamma} M, Y)$ is a support $\tau$-tilting $R$-module if and only if both $X_{\Lambda}$ and $Y_{\Gamma}$ are support $\tau$-tilting modules, $\operatorname{Hom}_{\Lambda}(Y \otimes_{\Gamma} M, \tau X) = 0$ and $\operatorname{Hom}_{\Lambda}(e\Lambda, Y \otimes_{\Gamma} M) = 0$ with $e$ the maximal idempotent such that $\operatorname{Hom}_{\Lambda}(e\Lambda,X) = 0$.