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On the relation between maximal rigid objects and $\tau $-tilting modules

Tom 142 / 2016

Pin Liu, Yunli Xie Colloquium Mathematicum 142 (2016), 169-178 MSC: 18E30, 16D90. DOI: 10.4064/cm142-2-2

Streszczenie

This note compares $\tau $-tilting modules and maximal rigid objects in the context of 2-Calabi–Yau triangulated categories. Let ${\mathcal C}$ be a 2-Calabi–Yau triangulated category with suspension functor $S$. Let $R$ be a maximal rigid object in ${\mathcal C}$ and let $\varGamma $ be the endomorphism algebra of $R$. Let $F$ be the functor $\operatorname {Hom}\nolimits _{{\mathcal C}}(R, -): {\mathcal C}\to \operatorname {mod}\nolimits \varGamma $. We prove that any $\tau $-tilting module over $\varGamma $ lifts uniquely to a maximal rigid object in ${\mathcal C}$ via $F$, and in turn, that projection from ${\mathcal C}$ to $\operatorname {mod}\nolimits \varGamma $ sends the maximal rigid objects which have no direct summands from $\operatorname {add}\nolimits SR$ to $\tau $-tilting $\varGamma $-modules, and in general, that the $\varGamma $-modules corresponding to the maximal rigid objects are the support $\tau $-tilting modules.

Autorzy

  • Pin LiuDepartment of Mathematics
    Southwest Jiaotong University
    611756 Chengdu, P.R. China
    e-mail
  • Yunli XieDepartment of Mathematics
    Southwest Jiaotong University
    611756 Chengdu, P.R. China
    e-mail

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