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 $K_{0}^{\text{sp}}({\mathcal{T}})$.

中文翻译：

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通过集群倾斜子类别划分三角类别的格罗腾迪克组

让$k$是一个场，让${\mathcal{C}}$做一个$k$-具有分裂幂等性的线性 Hom 有限三角范畴。在本文中，我们证明了在合适的情况下，格洛腾迪克群${\mathcal{C}}$，表示为$K_{0}({\mathcal{C}})$, 可以表示为更高聚类倾斜子类别的分裂格洛腾迪克群的商${\mathcal{C}}$. 我们证明的结果是 Xiao 和 Zhu 以及 Palu 对 Grothendieck 三角分类组的结果的更高版本。假使，假设$n\geqslant 2$是整数；${\mathcal{C}}$有一个 Serre 函子$\mathbb{S}$和$n$-集群倾斜子类别${\mathcal{T}}$这样$\operatorname{Ind}{\mathcal{T}}$是局部有界的。那么，对于每一个不可分解的$M$在${\mathcal{T}}$，有一个 Auslander-Reiten$(n+2)$- 角度${\mathcal{T}}$形式的$\mathbb{S}\unicode[STIX]{x1D6F4}^{-n}(M)\rightarrow T_{n-1}\rightarrow \cdots \rightarrow T_{0}\rightarrow M$和$$\begin{eqnarray}K_{0}({\mathcal{C}})\cong K_{0}^{\text{sp}}({\mathcal{T}})\left/\left\langle -[M]+(-1)^{n}[\mathbb{S}\unicode[STIX]{x1D6F4}^{-n}(M)]+\left.\mathop{\sum }_{i= 0}^{n-1}(-1)^{i}[T_{i}]\right|M\in \operatorname{Ind}{\mathcal{T}}\right\rangle .\right.\end {eqnarray}$$现在假设$d$是一个正整数并且${\mathcal{C}}$有个$d$-集群倾斜子类别${\mathcal{S}}$关闭$d$-暂停。然后，${\mathcal{S}}$是一个所谓的$(d+2)$-格罗腾迪克群的角度范畴$K_{0}({\mathcal{S}})$可以定义为某个商$K_{0}^{\text{sp}}({\mathcal{S}})$. 我们将展示$$\begin{eqnarray}K_{0}({\mathcal{C}})\cong K_{0}({\mathcal{S}}).\end{eqnarray}$$此外，假设$n=2d$，以上所有假设成立，并且${\mathcal{T}}\subseteq {\mathcal{S}}$. 那么我们的结果可以结合起来表达$K_{0}({\mathcal{S}})$作为商$K_{0}^{\text{sp}}({\mathcal{T}})$.