 
Summary: Jordan H¨older theorems for derived module categories of piecewise hereditary
algebras
Lidia Angeleri H¨ugel, Steffen Koenig, Qunhua Liu
Abstract. A Jordan H¨older theorem is established for derived module categories of piecewise
hereditary algebras, in particular for representations of quivers and for hereditary abelian cat
egories of a geometric nature. The resulting composition series of derived categories are shown
to be independent of the choice of bounded or unbounded derived module categories, and also
of the choice of finitely generated or arbitrary modules.
Introduction
Jordan H¨older theorems are classical and fundamental results in group theory and in module
theory. Under suitable assumptions, a Jordan H¨older theorem asserts the existence of a finite
`composition series', the subquotients of which are `simple' objects. A Jordan H¨older theorem
can be formulated when the concept of `short exact sequence' has been defined. Then an object
may be called simple if it is not the middle term of a short exact sequence, that is, it is not an
extension of another two objects in the given class of objects (groups, modules, . . . ). Then finite
series of short exact sequences can be considered, where the given object is the middle term of
the first sequence, the end terms of the first sequence are middle terms of further sequences, and
so on, until simple objects are reached and the process stops. A Jordan H¨older theorem states
finiteness of this process and the uniqueness of the simple constituents, up to a suitable notion
of isomorphism.
