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 Hom and 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 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 to a simple-minded collection iff the Ext-quiver of $\mathcal R$ is acyclic.