Jordan Hölder theorems for derived module categories of piecewise hereditary algebras

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

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 composi…

On the uniqueness of stratifications of derived module categories

Recollements of triangulated categories may be seen as exact sequences of such categories. Iterated recollements of triangulated categories are analogues of geometric or topological stratifications and of composition series of algebraic objects. We discuss the question of uniqueness of such a stratification, up to ordering and derived equivalence, for derived module categories. The main result is a positive answer in the form of a Jordan Hölder theorem for derived module categories of hereditary artin algebras. We also provide examples of derived simple rings.

精确结构和最大可 canonical Jordan 恢复子范畴对于类型 A 代数上的模的结构

Exact structures and maximal canonically Jordan recoverable subcategories for modules over type $A$ algebras

本文研究了类型 A 代数上模的精确结构和最大可 canonical Jordan 恢复性子分类，引入了 Gen-Sub 操作符并建立了其与 tilting 对象的关系。

根范畴的导出 Hall 代数

Derived Hall algebras of root categories

本文利用八面体公理定义了有限型遗传阿贝尔范畴根范畴的导出 Hall 代数，证明其同构于 Hall 代数的 Drinfeld double，并给出了 Laurent 对称函数环和双仿射 Hecke 代数的范畴化实现。