The Index With Respect to a Rigid Subcategory of a Triangulated Category

Abstract

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.