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

2014 ◽  
Vol 42 (10) ◽  
pp. 4296-4307 ◽  
Author(s):  
Pin Liu ◽  
Yunli Xie
Keyword(s):  
Triangulated Categories ◽  
Rigid Objects ◽  
Endomorphism Algebras

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.

scholarly journals Relative rigid objects in triangulated categories

2019 ◽  
Vol 520 ◽  
pp. 171-185 ◽  
Author(s):  
Changjian Fu ◽  
Shengfei Geng ◽  
Pin Liu
Keyword(s):  
Triangulated Categories ◽  
Rigid Objects

Transfer of derived equivalences from subalgebras to endomorphism algebras

2016 ◽  
Vol 15 (06) ◽  
pp. 1650100 ◽  
Author(s):  
Thomas Brüstle ◽  
Shengyong Pan
Keyword(s):  
Triangulated Categories ◽  
Derived Equivalences ◽  
Endomorphism Algebras ◽  
Yoneda Algebras

In this paper, we construct derived equivalences between subalgebras of some [Formula: see text]-Auslander–Yoneda algebras from a class of triangles in idempotent complete triangulated categories. The derived equivalences are obtained by transferring subalgebras induced by triangles to endomorphism algebras induced by approximation sequences.

scholarly journals RigidFusion: Robot Localisation and Mapping in Environments with Large Dynamic Rigid Objects

2021 ◽  
pp. 1-1
Author(s):  
Ran Long ◽  
Christian Rauch ◽  
Tianwei Zhang ◽  
Vladimir Ivan ◽  
Sethu Vijayakumar
Keyword(s):  
Rigid Objects

scholarly journals Recollements, comma categories and morphic enhancements

2021 ◽  
pp. 1-25
Author(s):  
Xiao-Wu Chen ◽  
Jue Le
Keyword(s):  
Module Category ◽  
Triangulated Category ◽  
Triangulated Categories ◽  
Comma Category ◽  
The Right ◽  
Projective Lines ◽  
Triangle Functor

For each recollement of triangulated categories, there is an epivalence between the middle category and the comma category associated with a triangle functor from the category on the right to the category on the left. For a morphic enhancement of a triangulated category $\mathcal {T}$ , there are three explicit ideals of the enhancing category, whose corresponding factor categories are all equivalent to the module category over $\mathcal {T}$ . Examples related to inflation categories and weighted projective lines are discussed.

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