Support $\tau$-Tilting Modules under Split-by-Nilpotent Extensions

Abstract

Let $\Gamma$ be a split extension of a finite-dimensional algebra $\Lambda$ by a nilpotent bimodule ${}_{\Lambda }{E}_{\Lambda }$, and let $(T,P)$ be a pair in $\mathrm{mod}\Lambda$ with $P$ projective. We prove that $(T{\otimes }_{\Lambda }{\Gamma }_{\Gamma },P{\otimes }_{\Lambda }{\Gamma }_{\Gamma })$ is a support $\tau$-tilting pair in $\mathrm{mod}\Gamma$ if and only if $(T,P)$ is a support $\tau$-tilting pair in $\mathrm{mod}\Lambda$ and $\Hom_Λ(T\otimes_ΛE,τT_Λ)=0=\Hom_Λ(P,T\otimes_ΛE)$. As applications, we obtain a necessary and sufficient condition such that $(T{\otimes }_{\Lambda }{\Gamma }_{\Gamma },P{\otimes }_{\Lambda }{\Gamma }_{\Gamma })$ is support $\tau$-tilting pair for a cluster-tilted algebra $\Gamma$ corresponding to a tilted algebra $\Lambda$; and we also get that if $T_1,T_2\in \mathrm{mod}\Lambda$ such that $T_1{\otimes }_{\Lambda }\Gamma$ and $T_2{\otimes }_{\Lambda }\Gamma$ are support $\tau$-tilting $\Gamma$-modules, then $T_1{\otimes }_{\Lambda }\Gamma$ is a left mutation of $T_2{\otimes }_{\Lambda }\Gamma$ if and only if $T_1$ is a left mutation of $T_2$.