Some geometric aspects of the calculus of fractions

1996 ◽  
Vol 4 (2-3) ◽  
pp. 139-165 ◽  
Author(s):  
Jean B�nabou
Keyword(s):  
Calculus Of Fractions

scholarly journals NEARLY MORITA EQUIVALENCES AND RIGID OBJECTS

2016 ◽  
Vol 225 ◽  
pp. 64-99 ◽  
Author(s):  
ROBERT J. MARSH ◽  
YANN PALU
Keyword(s):  
Simple Module ◽  
Triangulated Category ◽  
Triangulated Categories ◽  
Morita Equivalent ◽  
Rigid Objects ◽  
Endomorphism Algebras ◽  
Tilted Algebras ◽  
Calculus Of Fractions ◽  
Cluster Tilting ◽  
Maximal Rigid Object

If $T$ and $T^{\prime }$ are two cluster-tilting objects of an acyclic cluster category related by a mutation, their endomorphism algebras are nearly Morita equivalent (Buan et al., Cluster-tilted algebras, Trans. Amer. Math. Soc. 359(1) (2007), 323–332 (electronic)); that is, their module categories are equivalent “up to a simple module”. This result has been generalized by Yang, using a result of Plamondon, to any simple mutation of maximal rigid objects in a 2-Calabi–Yau triangulated category. In this paper, we investigate the more general case of any mutation of a (non-necessarily maximal) rigid object in a triangulated category with a Serre functor. In that setup, the endomorphism algebras might not be nearly Morita equivalent, and we obtain a weaker property that we call pseudo-Morita equivalence. Inspired by Buan and Marsh (From triangulated categories to module categories via localization II: calculus of fractions, J. Lond. Math. Soc. (2) 86(1) (2012), 152–170; From triangulated categories to module categories via localisation, Trans. Amer. Math. Soc. 365(6) (2013), 2845–2861), we also describe our result in terms of localizations.

Calculus of Fractions

Categories ◽  
1972 ◽  
pp. 256-291
Author(s):  
Horst Schubert
Keyword(s):  
Calculus Of Fractions
Sign in / Sign up

Export Citation Format

Share Document