Silting Modules over Triangular Matrix Rings

Abstract

Let $Λ,Γ$ be rings and $R=\left(\begin{array}{cc}Λ& 0 \\ M & Γ\end{array}\right)$ the triangular matrix ring with $M$ a $(Γ,Λ)$-bimodule. Let $X$ be a right $Λ$-module and $Y$ a right $Γ$-module. We prove that $(X, 0)$$\oplus$$(Y\otimes_ΓM, Y)$ is a silting right $R$-module if and only if both $X_Λ$ and $Y_Γ$ are silting modules and $Y\otimes_ΓM$ is generated by $X$. Furthermore, we prove that if $Λ$ and $Γ$ are finite dimensional algebras over an algebraically closed field and $X_Λ$ and $Y_Γ$ are finitely generated, then $(X, 0)$$\oplus$$(Y\otimes_ΓM, Y)$ is a support $τ$-tilting $R$-module if and only if both $X_Λ$ and $Y_Γ$ are support $τ$-tilting modules, $\Hom_Λ(Y\otimes_ΓM,τX)=0$ and $\Hom_Λ(eΛ, Y\otimes_ΓM)=0$ with $e$ the maximal idempotent such that $\Hom_Λ(eΛ, X)=0$.