Bulletin of the
Korean Mathematical Society BKMS

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-tilted algebras.

Keywords: Semibricks, support $\tau$-tilting modules, split-by-nilpotent extensions

MSC numbers: Primary 16G20, 16S90

Supported by: This work was partially supported by NSFC (No. 11971225)