Support $τ$-Tilting Modules under Split-by-Nilpotent Extensions

Let $Γ$ be a split extension of a finite-dimensional algebra $Λ$ by a nilpotent bimodule $_ΛE_Λ$, and let $(T,P)$ be a pair in $\modΛ$ with $P$ projective. We prove that $(T\otimes_ΛΓ_Γ, P\otimes_ΛΓ_Γ)$ is a support $τ$-tilting pair in $\mod Γ$ if and only if $(T,P)$ is a support $τ$-tilting pair in $\mod Λ$ 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 $(T\otimes_ΛΓ_Γ, P\otimes_ΛΓ_Γ)$ is support $τ$-tilting pair for a cluster-tilted algebra $Γ$ corresponding to a tilted algebra $Λ$; and we also get that if $T_1,T_2\in\modΛ$ such that $T_1\otimes_ΛΓ$ and $T_2\otimes_ΛΓ$ are support $τ$-tilting $Γ$-modules, then $T_1\otimes_ΛΓ$ is a left mutation of $T_2\otimes_ΛΓ$ if and only if $T_1$ is a left mutation of $T_2$.

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