(Contravariant) Koszul duality for DG algebras

TL;DR

This paper establishes a duality between derived categories of DG modules over certain DG algebras, generalizing Koszul duality to a broader class of algebras with specific homological properties.

Contribution

It proves a new form of Koszul duality for DG algebras with connected and degreewise finite homology, extending classical results to more general settings.

Findings

Derived categories of DG modules are equivalent under certain conditions.

The duality induces equivalences between categories of perfect modules.

Results apply to DG algebras with connected and simply connected homology.

Abstract

A DG algebras $A$ over a field $k$ with $H(A)$ connected and $H_{<0}(A)=0$ has a unique up to isomorphism DG module $K$ with $H(K)\cong k$. It is proved that if $H(A)$ is degreewise finite, then $RHom_A(?,K): D^{df}_{+}(A)^{op} \equiv D_{df}^{+}}(RHom_A(K,K))$ is an exact equivalence of derived categories of DG modules with degreewise finite-dimensional homology. It induces an equivalences of $D^{df}_{b}(A)^{op}$ and the category of perfect DG $RHom_A(K,K)$-modules, and vice-versa. Corresponding statements are proved also when $H(A)$ is simply connected and $H^{<0}(A)=0$.