The index with respect to a rigid subcategory of a triangulated category

TL;DR

This paper generalizes the concept of the index in triangulated categories by defining it relative to a contravariantly finite, rigid subcategory, extending previous work and analyzing its algebraic properties.

Contribution

It introduces a new index with respect to a rigid subcategory in triangulated categories and explores its properties and relations to Grothendieck groups.

Findings

Defines the index as a K_0-class in a relative extriangulated category.

Provides an additivity formula with an error term for the index on triangles.

Describes the structure of Grothendieck groups related to subcategories and the surjection between them.

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.