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

TL;DR

This paper investigates the structure of derived categories of quasi-hereditary algebras, showing they lack a unique composition series and answering open questions about their characterization.

Contribution

It demonstrates that having all factors as derived categories of vector spaces does not characterize quasi-hereditary algebras, and shows the absence of Jordan-H"older property.

Findings

Composition series with all factors as derived categories of vector spaces do not characterize quasi-hereditary algebras.

Derived categories of quasi-hereditary algebras can have multiple composition series of different lengths.

No Jordan-H"older property exists for these derived categories.

Abstract

We study composition series of derived module categories in the sense of Angeleri H\"ugel, K\"onig & 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 Bobi\'nski & 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-H\"older property for composition series of derived categories of quasi-hereditary algebras.