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

Abstract

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.

Spherical subcategories in representation theory

“…All previously known derived invariants for these algebras coincide. The non-equivalence of the derived categories was already shown independently by Bobiński in [4] and the second author in [27]. Moreover, examples in algebraic geometry suggest that these invariants carry geometric information: the behaviour under blow-ups is given in [21,Prop.…”

“…Since band modules have self-extensions, we know that H(F ) is a string module. However, the (−1)-extension of H(F ) forces a 1-extension; this can be shown along the lines of the proof of [27,Prop. 1.4].…”

“…That proof gives no direct anwer as to how the categories differ. In [27], the second author gives an independent proof of Proposition 1.2 which illuminates the difference between the categories somewhat. This inspired the use of spherelike objects shown here.…”

Examples of Finite Dimensional Algebras which do not Satisfy the Derived Jordan–Hölder Property

“…Fibonacci algebras [13,17] and their generalisations [16], then Λ(A, B) has finite global dimension and does not satisfy DJHP (Corollary 3.12). During the preparation of this paper, Martin Kalck informed us that he found a family of finite dimensional algebras of global dimension 2 (hence quasi-hereditary) for which DJHP fails (see [14,Proposition 3.4]); Changchang Xi informed us that [10, Theorem 1.1] can be used to construct finite dimensional algebras which do not satisfy DJHP. For two finite dimensional elementary algebras with the same number of isomorphism classes of simple modules, Chen and Xi constructed in the end of [10,Section 5] an upper triangular 2 × 2-matrix algebra, for which results similar to Theorem 3.8, (a) and (b) can be obtained.…”

In the bocs seat: Quasi-hereditary algebras and representation type

“…We stick to the one closest to exceptional collections in triangulated categories (which often arise in algebraic geometry). For other possibilities, see the three surveys mentioned in the beginning of this section as well as the articles by Kalck [46] and Krause [58] in this volume. (ii) Sometimes it is convenient to work with a partial order on {1, .…”