The index with respect to a rigid subcategory of a triangulated category (2201.00740v3)

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 in $\mathcal{C}$. We define the index $\mathrm{ind}{\mathcal{X}}(C)$ 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 $\mathrm{ind}{\mathcal{X}}$ 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 $\operatorname{Ker} Q$. We do this under certain assumptions.