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

TL;DR

This paper establishes a connection between silting objects, tilting objects, and simple-minded collections in the derived categories of hereditary categories, providing new proofs and criteria for their existence and completion.

Contribution

It proves equivalences between silting, tilting, and simple-minded collections in derived categories of hereditary categories and introduces new methods for their analysis.

Findings

Silting objects exist iff tilting objects exist in the derived category.

Every presilting object is a partial silting object.

A pre-simple-minded collection can be completed iff its Ext-quiver is acyclic.

Abstract

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.