A functorial approach to rank functions on triangulated categories

Abstract

We study rank functions on a triangulated category $\mathcal{C}$ via its abelianisation $\mathrm{mod}\mathcal{C}$. We prove that every rank function on $\mathcal{C}$ can be interpreted as an additive function on $\mathrm{mod}\mathcal{C}$. 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 $\mathrm{Mod}\mathcal{C}$. We study the connection between rank functions and functors from $\mathcal{C}$ to locally finite triangulated categories, generalising results by Chuang and Lazarev. In the special case $\mathcal{C}={\mathcal{T}}^c$ for a compactly generated triangulated category $\mathcal{T}$, this connection becomes particularly nice, providing a link between rank functions on $\mathcal{C}$ and smashing localisations of $\mathcal{T}$. In this context, any integral rank function can be described using the composition length with respect to certain endofinite objects in $\mathcal{T}$. Finally, if $\mathcal{C}=\mathrm{per}(A)$ for a differential graded algebra $A$, we classify homological epimorphisms $A\to B$ with $\mathrm{per}(B)$ locally finite via special rank functions which we call idempotent.

33 pages; v2: added details and references

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