On the existence of a derived equivalence between a Koszul algebra and its Yoneda algebra

TL;DR

This paper investigates conditions under which Koszul algebras are derived equivalent to their Yoneda algebras, establishing equivalences for simply connected and derived discrete cases, and exploring relations to hereditary algebras.

Contribution

It proves derived equivalences for simply connected Koszul algebras and characterizes when derived discrete Koszul algebras are equivalent to their Yoneda algebras, including cases related to hereditary algebras.

Findings

Simply connected Koszul algebras are derived equivalent to their Yoneda algebras.

Necessary and sufficient conditions for derived discrete Koszul algebras to be equivalent to their Yoneda algebras.

Characterization of derived equivalences between Koszul and hereditary algebras, especially in tame cases.

Abstract

In this paper we focus on the relations between the derived categories of a Koszul algebra and its Yoneda algebra, in particular we want to consider the cases where these categories are triangularly equivalent. We prove that the simply connected Koszul algebras are derived equivalent to their Yoneda algebras. We consider derived discrete Koszul algebras, and we give necessary and sufficient conditions for these Koszul algebras to be derived equivalent to their Yoneda algebras. Finally, we look at the Koszul algebras such that they are derived equivalent to a hereditary algebra. In the case that the hereditary algebra is tame, we characterize when these algebras are derived equivalent to their Yoneda algebras.