GROTHENDIECK GROUPS OF TRIANGULATED CATEGORIES VIA CLUSTER TILTING SUBCATEGORIES

• Published in: Nagoya mathematical journal Vol. 244; pp. 204 - 231
• Main Author: FEDELE, FRANCESCA
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.12.2021
• Subjects: Clusters; Integers; Quotients; 16E20; 18E30
• ISSN: 0027-7630; 2152-6842
• DOI: 10.1017/nmj.2020.12

Let $k$ be a field, and let ${\mathcal{C}}$ be a $k$ -linear, Hom-finite triangulated category with split idempotents. In this paper, we show that under suitable circumstances, the Grothendieck group of ${\mathcal{C}}$ , denoted by $K_{0}({\mathcal{C}})$ , can be expressed as a quotient of the split Grothendieck group of a higher cluster tilting subcategory of ${\mathcal{C}}$ . The results we prove are higher versions of results on Grothendieck groups of triangulated categories by Xiao and Zhu and by Palu. Assume that $n\geqslant 2$ is an integer; ${\mathcal{C}}$ has a Serre functor $\mathbb{S}$ and an $n$ -cluster tilting subcategory ${\mathcal{T}}$ such that $\operatorname{Ind}{\mathcal{T}}$ is locally bounded. Then, for every indecomposable $M$ in ${\mathcal{T}}$ , there is an Auslander–Reiten $(n+2)$ -angle in ${\mathcal{T}}$ of the form $\mathbb{S}\unicode[STIX]{x1D6F4}^{-n}(M)\rightarrow T_{n-1}\rightarrow \cdots \rightarrow T_{0}\rightarrow M$ and $$\begin{eqnarray}K_{0}({\mathcal{C}})\cong K_{0}^{\text{sp}}({\mathcal{T}})\left/\left\langle -[M]+(-1)^{n}[\mathbb{S}\unicode[STIX]{x1D6F4}^{-n}(M)]+\left.\mathop{\sum }_{i=0}^{n-1}(-1)^{i}[T_{i}]\right|M\in \operatorname{Ind}{\mathcal{T}}\right\rangle .\right.\end{eqnarray}$$ Assume now that $d$ is a positive integer and ${\mathcal{C}}$ has a $d$ -cluster tilting subcategory ${\mathcal{S}}$ closed under $d$ -suspension. Then, ${\mathcal{S}}$ is a so-called $(d+2)$ -angulated category whose Grothendieck group $K_{0}({\mathcal{S}})$ can be defined as a certain quotient of $K_{0}^{\text{sp}}({\mathcal{S}})$ . We will show $$\begin{eqnarray}K_{0}({\mathcal{C}})\cong K_{0}({\mathcal{S}}).\end{eqnarray}$$ Moreover, assume that $n=2d$ , that all the above assumptions hold, and that ${\mathcal{T}}\subseteq {\mathcal{S}}$ . Then our results can be combined to express $K_{0}({\mathcal{S}})$ as a quotient of $K_{0}^{\text{sp}}({\mathcal{T}})$ .