A functorial approach to rank functions on triangulated categories

Abstract

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