A functorial approach to rank functions on triangulated categories

<jats:p>We study rank functions on a triangulated category 𝒞 via its abelianisation <jats:inline-formula> <jats:alternatives> <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> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" xlink:href="graphic/j_crelle-2024-0009_ineq_0001. png" /> <jats:tex-math>\operatorname{mod}\mathcal{C}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. We prove that every rank function on 𝒞 can be interpreted as an additive function on <jats:inline-formula> <jats:alternatives> <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> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" xlink:href="graphic/j_crelle-2024-0009_ineq_0001. png" /> <jats:tex-math>\operatorname{mod}\mathcal{C}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. 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 <jats:inline-formula> <jats:alternatives> <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> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" xlink:href="graphic/j_crelle-2024-0009_ineq_0003. png" /> <jats:tex-math>\operatorname{Mod}\mathcal{C}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. 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 <jats:inline-formula> <jats:alternatives> <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> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" xlink:href="graphic/j_crelle-2024-0009_ineq_0004. png" /> <jats:tex-math>\mathcal{C}=\mathcal{T}^{c}</jats:tex-math> </jats:alternatives> </jats:inline-formula> 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 <jats:inline-formula> <jats:alternatives> <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> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" xlink:href="graphic/j_crelle-2024-0009_ineq_0005. png" /> <jats:tex-math>\mathcal{C}=\operatorname{per}(A)</jats:tex-math> </jats:alternatives> </jats:inline-formula> for a differential graded algebra 𝐴, we classify homological epimorphisms <jats:inline-formula> <jats:alternatives> <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> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" xlink:href="graphic/j_crelle-2024-0009_ineq_0006. png" /> <jats:tex-math>A\to B</jats:tex-math> </jats:alternatives> </jats:inline-formula> with <jats:inline-formula> <jats:alternatives> <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> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" xlink:href="graphic/j_crelle-2024-0009_ineq_0007. png" /> <jats:tex-math>\operatorname{per}(B)</jats:tex-math> </jats:alternatives> </jats:inline-formula> locally finite via special rank functions which we call idempotent. </jats:p>