On the relation between maximal rigid objects and τ-tilting modules

Pin Liu, and Yunli Xie. "On the relation between maximal rigid objects and τ-tilting modules." Colloquium Mathematicae 142.2 (2016): 169-178. <http://eudml.org/doc/286194>.

@article{PinLiu2016,
abstract = {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 $Hom_\{\}(R,-): → mod Γ$. 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.},
author = {Pin Liu, Yunli Xie},
journal = {Colloquium Mathematicae},
keywords = {$2$-Calabi-Yau category; cluster tilting object; $\tau $-tilting module; support $\tau $-tilting module; maximal rigid object},
language = {eng},
number = {2},
pages = {169-178},
title = {On the relation between maximal rigid objects and τ-tilting modules},
url = {http://eudml.org/doc/286194},
volume = {142},
year = {2016},
}

TY - JOUR
AU - Pin Liu
AU - Yunli Xie
TI - On the relation between maximal rigid objects and τ-tilting modules
JO - Colloquium Mathematicae
PY - 2016
VL - 142
IS - 2
SP - 169
EP - 178
AB - 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 $Hom_{}(R,-): → mod Γ$. 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.
LA - eng
KW - $2$-Calabi-Yau category; cluster tilting object; $\tau $-tilting module; support $\tau $-tilting module; maximal rigid object
UR - http://eudml.org/doc/286194
ER -