Derived categories of quasi-hereditary algebras and their derived composition series

Abstract

We study composition series of derived module categories in the sense of Angeleri Hügel, König & Liu for quasi-hereditary algebras. More precisely, we show that having a composition series with all factors being derived categories of vector spaces does not characterise derived categories of quasi-hereditay algebras. This gives a negative answer to a question of Liu & Yang and the proof also confirms part of a conjecture of Bobiński & Malicki. In another direction, we show that derived categories of quasi-hereditary algebras can have composition series with lots of different lengths and composition factors. In other words, there is no Jordan-Hölder property for composition series of derived categories of quasi-hereditary algebras.

36 pages, fixed an argument in proof of Prop. 1.4, results are unchanged, to appear in Proc. of Conference of the DFG priority program on Representation Theory, Bad Honnef, March 2015, comments are welcome!