On Modules over Endomorphism Algebras of Maximal Rigid Objects in 2-Calabi-Yau Triangulated Categories

Abstract

Abstract This note investigates the modules over the endomorphism algebras of maximal rigid objects in 2-Calabi-Yau triangulated categories. We study the possible complements for almost complete tilting modules. Combining with Happel's theorem, we show that the possible exchange sequences for tilting modules over such algebras are induced by the exchange triangles for maximal rigid objects in the corresponding 2-Calabi-Yau triangulated categories. For the modules of infinite projective dimension, we generalize a recent result by Beaudet–Brüstle–Todorov for cluster-tilted algebras. Key Words: 2-Calabi-Yau categoryMaximal rigid objectTilting module2010 Mathematics Subject Classification: 18E3016D90 ACKNOWLEDGMENTS The authors are grateful to the anonymous referee for the careful reading of the manuscript and very helpful advice.