A functorial approach to rank functions on triangulated categories

Abstract We study rank functions on a triangulated category 𝒞 via its abelianisation mod ⁡ C modC. We prove that every rank function on 𝒞 can be interpreted as an additive function on mod ⁡ C modC. 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 ModC. 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 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) C=per (A) for a differential graded algebra 𝐴, we classify homological epimorphisms A → B A B with per ⁡ (B) per (B) locally finite via special rank functions which we call idempotent.