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.

Categorical Koszul duality

“…There are also Koszul duality theorems involving coalgebras that are not necessarily conilpotent, cf. [14,7]. In this paper we investigate the analogues of the above-mentioned results in the context of global (i.e.…”

“…Our proof is constructed in such a way that it is suitable for generalization to homological algebra of the second kind. We review coderived categories of coalgebras and two versions of nonconilpotent Koszul duality between comodules over a (not necessarily conilpotent) dg coalgebra and modules over its Koszul dual dg algebra, following [22] and [14]. In particular, we recall the compactly generated derived category of the second kind D II c (A) for a dg algebra A, generated by its subcategory Perf II (A) of compact objects.…”

Differential graded Koszul duality: An introductory survey

“…This is one difference between the properties of the tensor algebras and the tensor coalgebras; this is also an explanation of the Kontsevich vanishing. 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].…”

“…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 $$A^{{\bullet }}$$ 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].…”