This note compares τ-tilting modules and maximal rigid objects in the context of 2-Calabi-Yau triangulated categories. Let be a 2-Calabi-Yau triangulated category with suspension functor S. Let R be a maximal rigid object in and let Γ be the endomorphism algebra of R. Let F be the functor . We prove that any τ-tilting module over Γ lifts uniquely to a maximal rigid object in via F, and in turn, that projection from to mod Γ sends the maximal rigid objects which have no direct summands from add SR to τ-tilting Γ-modules, and in general, that the Γ-modules corresponding to the maximal rigid objects are the support τ-tilting modules.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm142-2-2,
author = {Pin Liu and Yunli Xie},
title = {On the relation between maximal rigid objects and $\tau$-tilting modules},
journal = {Colloquium Mathematicae},
volume = {144},
year = {2016},
pages = {169-178},
zbl = {06498813},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm142-2-2}
}
Pin Liu; Yunli Xie. On the relation between maximal rigid objects and τ-tilting modules. Colloquium Mathematicae, Tome 144 (2016) pp. 169-178. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm142-2-2/