Sparseness of t‐structures and negative Calabi–Yau dimension in triangulated categories generated by a spherical object

Let k be an algebraically closed field and let ⊤ be the k‐linear algebraic triangulated category generated by a w‐spherical object for an integer w. For certain values of w this category is classic. For instance, if w = 0 then it is the compact derived category of the dual numbers over k. Our main results are that, for w ⩽ 0, the category ⊤ has no non‐trivial t‐structures, but does have one family of non‐trivial co‐t‐structures, whereas, for w ⩾ 1, the opposite statement holds. Moreover, without any claim to originality, we observe that for w ⩽ −1, the category ⊤ is a candidate to have negative Calabi–Yau dimension since Σw is the unique power of the suspension functor which is a Serre functor.

[1]  T. Holm,et al.  On a cluster category of infinite Dynkin type, and the relation to triangulations of the infinity-gon , 2012 .

[2]  Dong Yang,et al.  The Ringel–Hall Lie algebra of a spherical object , 2011, J. Lond. Math. Soc..

[3]  Osamu Iyama,et al.  Silting mutation in triangulated categories , 2010, J. Lond. Math. Soc..

[4]  Yu Zhou,et al.  Mutation of Torsion Pairs in Triangulated Categories and its Geometric Realization , 2011, 1105.3521.

[5]  M. Bondarko.  Weight structures vs. $t$-structures; weight filtrations, spectral sequences, and complexes (for motives and in general) , 2007, 0704.4003.

[6]  B. Keller,et al.  Deformed Calabi–Yau completions , 2009, 0908.3499.

[7]  B. Keller,et al.  Derived equivalences from mutations of quivers with potential , 2009, 0906.0761.

[8]  L. Positselski.  Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence , 2009, 0905.2621.

[9]  B. Keller,et al.  The Hall algebra of a spherical object , 2008, 0810.5546.

[10]  Claire Amiot.  Cluster categories for algebras of global dimension 2 and quivers with potential , 2008, 0805.1035.

[11]  Y. Yoshino,et al.  Mutation in triangulated categories and rigid Cohen–Macaulay modules , 2006, math/0607736.

[12]  Idun Reiten,et al.  Homological and Homotopical Aspects of Torsion Theories , 2007 .

[13]  David Pauksztello.  Compact corigid objects in triangulated categories and co-t-structures , 2007, 0705.0102.

[14]  Peter Jørgensen.  Auslander-Reiten theory over topological spaces , 2004 .

[15]  M. Hoshino,et al.  On t-structures and torsion theories induced by compact objects , 2000, math/0005172.

[16]  Jun-ichi Miyachi,et al.  Localization of triangulated categories and derived categories , 1991 .

[17]  Dieter Happel,et al.  On the derived category of a finite-dimensional algebra , 1987 .

[18]  K. Bongartz.  Algebras and Quadratic Forms , 1983 .