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

Pin Liu ¹   ; Yunli Xie ¹

¹ Department of Mathematics Southwest Jiaotong University 611756 Chengdu, P.R. China

Colloquium Mathematicum, Tome 142 (2016) no. 2, pp. 169-178

Résumé

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.

Détail

DOI : 10.4064/cm142-2-2

Keywords: note compares tau tilting modules maximal rigid objects context calabi yau triangulated categories mathcal calabi yau triangulated category suspension functor maximal rigid object mathcal vargamma endomorphism algebra functor operatorname hom nolimits mathcal mathcal operatorname mod nolimits vargamma prove tau tilting module vargamma lifts uniquely maximal rigid object mathcal via nbsp turn projection mathcal operatorname mod nolimits vargamma sends maximal rigid objects which have direct summands operatorname nolimits tau tilting vargamma modules general vargamma modules corresponding maximal rigid objects support tau tilting modules