On the relation between maximal rigid objects and $\tau $-tilting modules
Tom 142 / 2016
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.
