Contravariant Koszul duality between non-positive and positive dg algebras

TL;DR

This paper explores the duality between non-positive and positive dg algebras via Koszul duality, characterizing conditions for local finiteness and establishing equivalences between derived categories.

Contribution

It provides a characterization of locally finite positive dg algebras with locally finite Koszul duals and demonstrates contravariant equivalences between key derived categories.

Findings

Characterization of locally finite positive dg algebras with locally finite Koszul duals

Contravariant equivalences between perfect and perfectly valued derived categories

Establishment of an ST-correspondence and properties of hearts in triangulated categories

Abstract

The Koszul dual of locally finite non-positive dg algebra is locally finite positive dg algebra. However, the Koszul dual of locally finite positive dg algebra is not necessary locally finite. We characterize locally finite positive dg algebras whose Koszul dual is locally finite. Moreover, we show that the Koszul dual functor induces contravariant equivalences between the perfect derived category and the perfectly valued derived category. As an application of Koszul dualities, we establish an ST-correspondence. We also show that, under some assumption, every covariantly finite bounded heart is a length heart, and the triangulated analogy of Smal{\o}'s symmetry holds.