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December, 2020 Silting Modules over Triangular Matrix Rings
Hanpeng Gao, Zhaoyong Huang
Taiwanese J. Math. 24(6): 1417-1437 (December, 2020). DOI: 10.11650/tjm/200204


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 $\operatorname{Hom}_{\Lambda}(e\Lambda,X) = 0$.


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Hanpeng Gao. Zhaoyong Huang. "Silting Modules over Triangular Matrix Rings." Taiwanese J. Math. 24 (6) 1417 - 1437, December, 2020.


Received: 3 December 2019; Revised: 10 February 2020; Accepted: 16 February 2020; Published: December, 2020
First available in Project Euclid: 19 November 2020

MathSciNet: MR4176880
Digital Object Identifier: 10.11650/tjm/200204

Primary: 16E30 , 16G10

Keywords: $\tau$-rigid modules , (partial) silting modules , support $\tau$-tilting modules , tilting modules , triangular matrix rings

Rights: Copyright © 2020 The Mathematical Society of the Republic of China


Vol.24 • No. 6 • December, 2020
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