A functorial approach to rank functions on triangulated categories

We study rank functions on a triangulated category 𝒞 via its abelianisation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>mod</m:mi> <m:mo lspace="0.167em">⁡</m:mo> <m:mi mathvariant="script">C</m:mi> </m:mrow> </m:math> modC . We prove that every rank function on 𝒞 can be interpreted as an additive function on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>mod</m:mi> <m:mo lspace="0.167em">⁡</m:mo> <m:mi mathvariant="script">C</m:mi> </m:mrow> </m:math> 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Mod</m:mi> <m:mo lspace="0.167em">⁡</m:mo> <m:mi mathvariant="script">C</m:mi> </m:mrow> </m:math> 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="script">C</m:mi> <m:mo>=</m:mo> <m:msup> <m:mi mathvariant="script">T</m:mi> <m:mi>c</m:mi> </m:msup> </m:mrow> </m:math> C=Tᶜ 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="script">C</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mi>per</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>A</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> C=per(A) for a differential graded algebra 𝐴, we classify homological epimorphisms <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>A</m:mi> <m:mo stretchy="false">→</m:mo> <m:mi>B</m:mi> </m:mrow> </m:math> A→ B with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>per</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>B</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> per(B) locally finite via special rank functions which we call idempotent.