TRIANGULATED CATEGORIES WITH CLUSTER TILTING SUBCATEGORIES
TRIANGULATED CATEGORIES WITH CLUSTER TILTING SUBCATEGORIES
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具有集群倾斜子类别的三角类别
DOI：
10.2140/pjm.2019.301.703
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发表时间：
2019
影响因子：
0.6
通讯作者：
Zhu Bin
中科院分区：
文献类型：
--
作者：
Yang Wuzhong;Zhou Panyue;Zhu Bin
设$\C$是一个具有簇倾斜子范畴$\T$的三角范畴。我们引入了$\C$的$\T[1]$-簇倾斜子范畴(也称为幽灵簇倾斜子范畴)的概念，它是簇倾斜子范畴的推广。我们首先发展了鬼簇倾斜子范畴的基本理论。其次，我们研究了鬼团倾斜理论与$-倾斜理论之间的联系：受Iyama，Jorgensen和Yang\cite{ijy}工作的启发，我们引入了$\tau-倾斜子范畴和$\mod\T$的倾斜子范畴的概念。我们证明了$C$的弱$T[1]$簇倾斜子范畴与$mod T$的支集$tau$倾斜子范畴之间存在双射。此外，我们还计算出了$\mod\T$中对应于$\C$的簇倾斜子范畴的子范畴。这推广和改进了Adachi-Iyama-Reiten\cite{AIR}，Beligiannis\cite{Be2}和Yang-朱\cite{YZ}的几个结果。最后，我们证明了虚星团倾斜体的定义与第一和第三作者在Cite{YZ}中引入的相对星团倾斜体的定义是等价的。
Let $\C$ be a triangulated category with a cluster tilting subcategory $\T$. We introduce the notion of $\T[1]$-cluster tilting subcategories (also called ghost cluster tilting subcategories) of $\C$, which are a generalization of cluster tilting subcategories. We first develop a basic theory on ghost cluster tilting subcategories. Secondly, we study links between ghost cluster tilting theory and $\tau$-tilting theory: Inspired by the work of Iyama, Jorgensen and Yang \cite{ijy}, we introduce the notion of $\tau$-tilting subcategories and tilting subcategories of $\mod\T$. We show that there exists a bijection between weak $\T[1]$-cluster tilting subcategories of $\C$ and support $\tau$-tilting subcategories of $\mod\T$. Moreover, we figure out the subcategories of $\mod\T$ which correspond to cluster tilting subcategories of $\C$. This generalizes and improves several results by Adachi-Iyama-Reiten \cite{AIR}, Beligiannis \cite{Be2}, and Yang-Zhu \cite{YZ}. Finally, we prove that the definition of ghost cluster tilting objects is equivalent to the definition of relative cluster tilting objects introduced by the first and the third author in \cite{YZ}.