A reduction approach to silting objects for derived categories of hereditary categories

Let $mathcal{H}$ be a hereditary abelian category over a field $k$ with finite dimensional $operatorname{Hom}$ and $operatorname{Ext}$ spaces. It is proved that the bounded derived category $mathcal{D}^b(mathcal{H})$ has a silting object iff $mathcal{H}$ has a tilting object iff $mathcal{D}^b(mathcal{H})$ has a simple-minded collection with acyclic $operatorname{Ext}$-quiver. Along the way, we obtain a new proof for the fact that every presilting object of $mathcal{D}^b(mathcal{H})$ is a partial silting object. We also consider the question of complements for pre-simple-minded collections. In contrast to presilting objects, a pre-simple-minded collection $mathcal{R}$ of $mathcal{D}^b(mathcal{H})$ can be completed into a simple-minded collection iff the $operatorname{Ext}$-quiver of $mathcal{R}$ is acyclic.