Silting Modules over Triangular Matrix Rings

Abstract

Let $\Lambda ,\Gamma$ be rings and $R=\left(\begin{array}{cc}\Lambda & 0\\ M& \Gamma \end{array}\right)$ 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, $\Hom_Λ(Y\otimes_ΓM,τX)=0$ and $\Hom_Λ(eΛ, Y\otimes_ΓM)=0$ with $e$ the maximal idempotent such that $\Hom_Λ(eΛ, X)=0$.

17 pages, accepted for publication in Taiwanese Journal of Mathematics