On the existence of cluster tilting objects in triangulated categories
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 general local complete intersection rings. By using the support theory of \textit{L. L. Avramov} and \textit{R.-O. Buchweitz} [J. Algebra 230, No. 1, 24--67 (2000; Zbl 1011.13007)], he shows that the triangulated subcategory of objects of complexity at most~\(t\), for \(1\leq t < c\) where \(c\) denotes the codimension of the complete intersection, does not admit any cluster tilting object.
scientific article; zbMATH DE number 6339745
scientific article