25 October 2012
A triangulated category admits a strong generator if, roughly speaking, every object can be built in a globally bounded number of steps starting from a single object and taking iterated cones. The importance of strong generators was demonstrated by Bondal and van den Bergh, who proved that the existence of such objects often gives you a representability theorem for cohomological functors. The importance was further emphasised by Rouquier, who introduced the dimension of triangulated categories, and tied this numerical invariant to the representation dimension. In this talk I will discuss the generation time for strong generators (the least number of cones required to build every object in the category) and a refinement of the dimension which is due to Orlov: the set of all integers that occur as a generation time. After introducing the necessary terminology, I will focus on categories occurring in representation theory and explain how to compute this invariant for the bounded derived category of the path algebras of type A and D, as well as the corresponding cluster categories.
- Representation Theory Seminar