Year: 2023
Type: article
Abstract: Palu defined the index with respect to a cluster tilting object in a suitable triangulated category, in order to better understand the Caldero–Chapoton map that exhibits the connection between cluster algebras and representation theory. We push this further by proposing an index with respect to a contravariantly finite, rigid subcategory, and we show this index behaves similarly to the classical index. Let ${\mathcal{C}}$ be a skeletally small triangulated category with split idempotents, which is thus an extriangulated category $({\mathcal{C}},{\mathbb{E}},{\mathfrak{s}})$. Suppose ${\mathcal{X}}$ is a contravariantly finite, rigid subcategory of ${\mathcal{C}}$. We define the index $ {\operatorname{\textrm{ind}}}{_{{\mathcal{X}}}}(C)$ of an object $C\in{\mathcal{C}}$ with respect to ${\mathcal{X}}$ as the $ {K}{_{0}}$-class $[C {]}{_{{\mathcal{X}}}}$ in Grothendieck group $ {K}{_{0}}({\mathcal{C}}, {{\mathbb{E}}}{_{{\mathcal{X}}}}, {{\mathfrak{s}}}{_{{\mathcal{X}}}})$ of the relative extriangulated category $({\mathcal{C}}, {{\mathbb{E}}}{_{{\mathcal{X}}}}, {{\mathfrak{s}}}{_{{\mathcal{X}}}})$. By analogy to the classical case, we give an additivity formula with error term for $ {\operatorname{\textrm{ind}}}{_{{\mathcal{X}}}}$ on triangles in ${\mathcal{C}}$. In case ${\mathcal{X}}$ is contained in another suitable subcategory ${\mathcal{T}}$ of ${\mathcal{C}}$, there is a surjection $Q\colon{K}{_{0}}({\mathcal{C}}, {{\mathbb{E}}}{_{{\mathcal{T}}}}, {{\mathfrak{s}}}{_{{\mathcal{T}}}}) \twoheadrightarrow{K}{_{0}}({\mathcal{C}}, {{\mathbb{E}}}{_{{\mathcal{X}}}}, {{\mathfrak{s}}}{_{{\mathcal{X}}}})$. Thus, in order to describe $ {K}{_{0}}({\mathcal{C}}, {{\mathbb{E}}}{_{{\mathcal{X}}}}, {{\mathfrak{s}}}{_{{\mathcal{X}}}})$, it suffices to determine $ {K}{_{0}}({\mathcal{C}}, {{\mathbb{E}}}{_{{\mathcal{T}}}}, {{\mathfrak{s}}}{_{{\mathcal{T}}}})$ and $\operatorname{Ker}\nolimits Q$. We do this under certain assumptions.
DOI: 10.1093/imrn/rnad130
Authors: Peter Jørgensen, Amit Shah
Institution: Aarhus University
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4384492739