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 $\mathrm{Hom}$ and $\mathrm{Ext}$ 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 $\mathrm{Ext}$-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 $\mathrm{Ext}$-quiver of $\mathcal{R}$ is acyclic.