PUB - Publikationen an der Universität Bielefeld

We study rank functions on a triangulated category C via its abelianisation mod C. We prove that every rank function on C can be interpreted as an additive function on mod C. As a consequence, every integral rank function has a unique decomposition into irreducible ones. Furthermore, we relate integral rank functions to a number of important concepts in the functor category Mod C. We study the connection between rank functions and functors from C to locally finite triangulated categories, generalising results by Chuang and Lazarev. In the special case C = T-c for a compactly generated triangulated category T, this connection becomes particularly nice, providing a link between rank functions on C and smashing localisations of T. In this context, any integral rank function can be described using the composition length with respect to certain endofinite objects in T. Finally, if C = per (A) for a differential graded algebra A, we classify homological epimorphisms A -> B with per (B) locally finite via special rank functions which we call idempotent.