Koszul duality for compactly generated derived categories of second kind

TL;DR

This paper develops a new model category structure for dg modules over any dg algebra, establishing a duality with comodules over an extended bar construction, thus broadening the scope of dg Koszul duality.

Contribution

It introduces a closed model category on dg modules with a homotopy category generated by finitely generated free modules and proves its Quillen equivalence to comodules over an extended bar construction.

Findings

Constructed a closed model category structure on dg modules.

Proved Quillen equivalence with comodules over an extended bar construction.

Generalized dg Koszul duality for associative algebras.

Abstract

For any dg algebra $A$ we construct a closed model category structure on dg $A$-modules such that the corresponding homotopy category is compactly generated by dg $A$-modules that are finitely generated and free over $A$ (disregarding the differential). We prove that this closed model category is Quillen equivalent to the category of comodules over a certain, possibly nonconilpotent dg coalgebra, a so-called extended bar construction of $A$. This generalises and complements certain aspects of dg Koszul duality for associative algebras.