Non-semistable exceptional objects in hereditary categories

Abstract

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.