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Support $\tau $-tilting modules under split-by-nilpotent extensions

Volume 160 / 2020

Hanpeng Gao, Zhaoyong Huang Colloquium Mathematicum 160 (2020), 247-262 MSC: 16G20, 16E30. DOI: 10.4064/cm7774-3-2019 Published online: 6 February 2020

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 $\operatorname{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 $\operatorname{mod} \Gamma $ if and only if $(T,P)$ is a support $\tau $-tilting pair in $\operatorname{mod} \Lambda $ and $\operatorname{Hom} _\Lambda (T\otimes _\Lambda E,\tau T_\Lambda )=0=\operatorname{Hom} _\Lambda (P,T\otimes _\Lambda E)$. As applications, we obtain a necessary and sufficient condition for $(T\otimes _\Lambda \Gamma _\Gamma , P\otimes _\Lambda \Gamma _\Gamma )$ to be a support $\tau $-tilting pair for a cluster-tilted algebra $\Gamma $ corresponding to a tilted algebra $\Lambda $; and we also show that if $T_1,T_2\in \operatorname{mod} \Lambda $ are 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 $T_2$.

Authors

  • Hanpeng GaoDepartment of Mathematics
    Nanjing University
    210093 Nanjing, Juangsu, P.R. China
    e-mail
  • Zhaoyong HuangDepartment of Mathematics
    Nanjing University
    210093 Nanjing, Juangsu, P.R. China
    e-mail

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