On the existence of cluster tilting objects in triangulated categories
type: Item
instance of: scholarly article
title: On the existence of cluster tilting objects in triangulated categories
zbMATH Open document ID: 1309.13015
author: Petter Andreas Bergh
published in: Journal of Algebra
publication date: 2014-09-04T00:00:00Z
reviewed by: Ivo Dell'Ambrogio
Mathematics Subject Classification ID: 13D02
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 16E05
Mathematics Subject Classification ID: 16E30
Mathematics Subject Classification ID: 18E30
zbMATH DE Number: 6339745
zbMATH Keywords: cluster tilting objects
zbMATH Keywords: complexity
zbMATH Keywords: local Gorenstein algebras
zbMATH Keywords: triangulated categories
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2014.06.024
OpenAlex ID: W2093043423
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DOI: 10.1016/J.JALGEBRA.2014.06.024
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review text: The present article investigates the existence of cluster tilting objects in triangulated categories. The main thrust is that the existence of such objects forces bounds in \textit{complexity}, in the general sense of \textit{P. A. Bergh} and \textit{S. Oppermann} [Int. Math. Res. Not. 2011, No. 22, 5241--5258 (2011; Zbl 1253.16009)].NEWLINENEWLINEMost results concern triangulated categories that are Hom-finite over a field, are Krull-Schmidt, and have Serre duality. The idea is that every indecomposable summand of a cluster tilting object must be periodic for some combination of the suspension and the Serre functor.NEWLINENEWLINEThe first main result (Theorem 2.2) concerns the stable module category of a self-injective algebra: if it contains an \(n\)-cluster tilting object, then the positive and negative complexity of any two objects is at most one. One can deduce from this that the complexity (in the usual sense of the growth of a minimal projective resolution) of any finitely generated module is at most one, which is a result of \textit{K. Erdmann} and \textit{T. Holm} [Proc. Am. Math. Soc. 136, No. 9, 3069--3078 (2008; Zbl 1153.16010)].NEWLINENEWLINEThe second main result (Theorem 3.3) concerns \(d\)-Calabi-Yau categories: if they contain an \(n\)-cluster tilting object and either \(n \neq d\) or \(n=d=1\), then again the complexity must be bounded by one. (This is false for \(n = d \neq 1\).)NEWLINENEWLINEExamples of this situation are provided by the singularity category (i.e., the stable category of maximal Cohen-Macaulay modules) of isolated singularity local rings. The author deduces from the latter complexity bound that in almost all cases -- when comparing \(n\) with the Calabi-Yau dimension~\(d\) -- if an \(n\)-cluster tilting object exists then the singularity is necessarily a hypersurface.NEWLINENEWLINEFinally, the author leaves the setting of Krull-Schmidt Hom-finite categories in order to prove something about the singularity category of more genera…
description: scientific article; zbMATH DE number 6339745