Non-semistable exceptional objects in hereditary categories
Non-semistable exceptional objects in hereditary categories
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DOI：
10.1093/imrn/rnv336
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发表时间：
2013-11
期刊：
影响因子：
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通讯作者：
G. Dimitrov;L. Katzarkov
中科院分区：
文献类型：
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作者：
G. Dimitrov;L. Katzarkov
对于三角分类上给定的稳定性条件$\sigma$，我们将$\sigma$ -异常集合定义为ext -异常集合，其元素是$\sigma$ -半稳定的，其相位包含在长度为1的开放区间内。如果存在一个完整的$\sigma$ -异常集合，那么$\sigma$将由该集合在E. Macrì描述的过程中生成。从一个非半稳定异常对象构造长度至少为$D^b(\mathcal A)$中3的$\sigma$ -异常集合，其中$\mathcal A$是一个遗传的半有限阿贝尔范畴，我们引入了ext -非平凡对（异常对象$X,Y\in \mathcal A$与${\rm Ext}^1(X,Y)\neq 0$和${\rm Ext}^1(Y,X)\neq 0$的对）的某些条件。在详细研究了两个分别具有三个顶点和四个顶点的温和颤抖$Q_1$和$Q_2$的特殊对象之后，我们观察到在$Rep_k(Q_1)$, $Rep_k(Q_2)$中确实存在所需的条件，其中$k$是一个代数封闭场。结合这些发现，我们证明了对于每个$\sigma\in {\rm Stab}(D^b(Q_1))$存在一个完整的$\sigma$ -例外集合。由此可见，${\rm Stab}(D^b(Q_1))$是相通的。
For a given stability condition $\sigma$ on a triangulated category we define a $\sigma$-exceptional collection as an Ext-exceptional collection, whose elements are $\sigma$-semistable with phases contained in an open interval of length one. If there exists a full $\sigma$-exceptional collection, then $\sigma$ is generated by this collection in a procedure described by E. Macr\`i. Constructing $\sigma$-exceptional collections of length at least three in $D^b(\mathcal A)$ from a non-semistable exceptional object, where $\mathcal A$ is a hereditary hom-finite abelian category, we introduce certain conditions on the Ext-nontrivial couples (couples of exceptional objects $X,Y\in \mathcal A$ with ${\rm Ext}^1(X,Y)\neq 0$ and ${\rm Ext}^1(Y,X)\neq 0$). After a detailed study of the exceptional objects of two tame quivers $Q_1$ and $Q_2$ with three and four vertices, respectively, we observe that the needed conditions do hold in $Rep_k(Q_1)$, $Rep_k(Q_2)$, where $k$ is an algebraically closed field. Combining these findings, we prove that for each $\sigma\in {\rm Stab}(D^b(Q_1))$ there exists a full $\sigma$-exceptional collection. It follows that ${\rm Stab}(D^b(Q_1))$ is connected.