Semibricks over split-by-nilpotent extensions
Bull. Korean Math. Soc.
Published online September 3, 2020
Hanpeng Gao
Nanjing University
Abstract : In this paper, we prove that there is a bijection between the $\tau$-tilting modules and the
sincere left finite semibricks. We also construct (sincere) semibricks over split-by-nilpotent extensions. More precisely, let $\Gamma$ be a split-by-nilpotent extension of a finite-dimensional algebra $\Lambda$ by a nilpotent bimodule $_\Lambda E_\Lambda$, and $\mathcal{S}\subseteq\mod\Lambda$. We prove that $\mathcal{S}\otimes_\Lambda\Gamma$ is a (sincere) semibrick in $\mod\Gamma$ if and only if
$\mathcal{S}$ is a semibrick in $\mod\Lambda$ and $Hom_\Lambda(\mathcal{S},\mathcal{S}\otimes_\Lambda E)=0$ (and $\mathcal{S}\cup\mathcal{S}\otimes_\Lambda E$ is sincere). As an application, we can construct $\tau$-tilting modules and support $\tau$-tilting modules over $\tau$-tilting finite cluster algebras.
Keywords : Semibricks, support $\tau$-tilting modules, split-by-nilpotent extensions
MSC numbers : 16G20, 16S90
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