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

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 $Q:K_0(\mathcal{C},{\mathbb{E}}_{\mathcal{T}},{\mathfrak{s}}_{\mathcal{T}})\twoheadrightarrow K_0(\mathcal{C},{\mathbb{E}}_{\mathcal{X}},{\mathfrak{s}}_{\mathcal{X}})$. 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 $\mathrm{Ker}Q$. We do this under certain assumptions.

V1: 26 pages. V2: 26 pages, minor title change, added to Remark 4.13. V3: Several changes made following a review. We now use the notation add(X*Y) for the extension subcategory previously denoted X*Y

References in corpus (3)

• The category of extensions and a characterisation of $n$-exangulated functors
• Integral and quasi-abelian hearts of twin cotorsion pairs on extriangulated categories
• Grothendieck groups of $d$-exangulated categories and a modified Caldero-Chapoton map

Cited by in corpus (6)

• The category of extensions and a characterisation of $n$-exangulated functors
• A resolution theorem for extriangulated categories with applications to the index
• Grothendieck groups of $d$-exangulated categories and a modified Caldero-Chapoton map
• The index in $d$-exact categories
• Localization of triangulated categories with respect to extension-closed subcategories
• Stratifying systems and Jordan-Hölder extriangulated categories