Koszul duality for compactly generated derived categories of second kind

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.

Differential graded Koszul duality: An introductory survey

“…The problem of nonfunctoriality of the bar-construction with respect to change-of-connection morphisms of CDG-algebras is resolved by passing from the usual (conilpotent) to the 'extended' (nonconilpotent) bar-construction of [28,Definition 2.5]. See the discussion in [28, section 4].…”

“…A version of nonconilpotent Koszul duality (Theorem 6.13) with the roles of the algebras and coalgebras switched (starting from a DG-algebra 𝐴 • and considering the cofree nonconilpotent coalgebra spanned by 𝐴 + , or rather, the vector space dual topological algebra) was suggested in the paper [28]. For a version of derived nonhomogeneous Koszul duality for DG-categories, see [32].…”