Koszul duality for compactly generated derived categories of second kind

Ai Guan,Andrey Lazarev

doi:10.4171/jncg/438

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.

Highlights

Koszul duality is an algebraic phenomenon that goes back to Quillen’s work [Qui69] on rational homotopy theory; it later manifested itself in many different contexts: operads [GK94], deformation theory [Hin01], representation theory [BGS96] and numerous others
The modern understanding of Koszul duality for differential graded algebras and dg modules has been formulated in [Pos11]. According to this formulation there is an adjunction between the categories of augmented dg algebras and conilpotent dg coalgebras, given by bar and cobar constructions, which becomes a Quillen equivalence under certain model category structures
The conilpotent dg coalgebra associated to an augmented dg algebra by this equivalence is called its Koszul dual; the augmented dg algebra associated to a conilpotent dg coalgebra is called its Koszul dual

Summary

Introduction

Koszul duality is an algebraic phenomenon that goes back to Quillen’s work [Qui69] on rational homotopy theory; it later manifested itself in many different contexts: operads [GK94], deformation theory [Hin01], representation theory [BGS96] and numerous others. The modern understanding of Koszul duality for differential graded (dg) algebras and dg modules has been formulated in [Pos11] According to this formulation there is an adjunction between the categories of augmented dg algebras and conilpotent dg coalgebras, given by bar and cobar constructions, which becomes a Quillen equivalence under certain model category structures. Positselski proves ([Pos, Theorem 6.7]) that there is a Koszul duality between dg comodules over a possibly nonconilpotent dg coalgebra and modules over its Koszul dual dg algebra This time both closed model structures are of the second kind: the weak equivalences on dg modules are not merely. In the present paper we construct a complementary version of Positselski’s non-conilpotent Koszul duality as a Quillen equivalence between closed model categories of dg modules over a dg algebra and comodules over its “Koszul dual” dg coalgebra. The authors would like to thank Leonid Positselski for freely sharing his expert knowledge of the subject matter and Joe Chuang for many stimulating discussions

Extended bar construction

Koszul duality for modules

Curved Koszul duality for modules