论文标题
支持$τ$ - 在逐型扩展下的模块
Support $τ$-Tilting Modules under Split-by-Nilpotent Extensions
论文作者
论文摘要
令$γ$为nilpotent bimodule $_λe_λ$的有限维代数$λ$的分式扩展，然后让$（t，p）$为$ \modλ$，带有$ p $ poxtibive。我们证明$（t \otimes_λγ_γ，p \otimes_λγ_γ）$是支持$τ$ - 以$ \ modγγ$ in $ \ modγ$，并且仅当$（t，p）$是$ \modλ$和$ \modλ$和us $ \hom_λ（t \otimes_λe，τt_λ）= 0 = \hom_λ（p，t \otimes_λe）$。作为应用程序，我们获得了必要且充分的条件，因此$（t \otimes_λγ_γ，p \otimes_λγ_γ）$是支持$τ$ - 用于聚类倾斜的algebra $γ$，对应于倾斜的algebra $λ$;而且，我们还会明白，如果$ t_1，t_2 \ in \modλ$，以至于$ t_1 \otimes_λγ$和$ t_2 \otimes_λγ$是支持$τ$ - 利用$γ$ -MODULES，那么$ t_1 \otimes_λγ$是$ t_2 $ t_2 $ t_2 $ ift $ $ T_2 $。
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 $T_2$.
