A New Model for Compactly Generated Derived Categories of the Second Kind and Curved Koszul Triality
Yannick Hoyer, Kristoffer Rank Rasmussen
TL;DR
The paper extends Koszul duality to curved, non-conilpotent settings by constructing an injective Guan–Lazarev model structure $A- mod^{II}_{ctr}$ on CDG $A$-modules via the extended bar construction $\check{B}A$ and the twisting $\tau$. It proves a Quillen equivalence between $A- mod^{II}_{ctr}$ and the contraderived model category $D^{ctr}(\check{B}A- contra)$, thereby connecting curved Koszul duality with Guan–Lazarev’s framework and Positselski’s comodule–contramodule correspondence. The curved Koszul triality is established through a commutative triangle of triangulated equivalences among $D_c^{II}(A- mod)$, $D^{co}(\check{B}A- comod)$, and $D^{ctr}(\check{B}A- contra)$, with a two-variable tensor–Hom adjunction detailed for these second-kind model structures. This work enables a robust derived-category-theoretic treatment of curved Koszul duality and demonstrates that tensor products behave as Quillen bifunctors in this setting, providing tools for compactly generated derived categories of the second kind in curved contexts.
Abstract
For any curved differential graded algebra $A$, we define a new model structure on the category of curved differential graded $A$-modules, called the injective Guan-Lazarev model structure. We prove that the category of CDG $A$-modules with this model structure is Quillen equivalent to the category of curved differential graded contramodules over the extended bar-construction of $A$, equipped with the contraderived model structure. This result can be seen as bridging the gap between Positselski's theory of conilpotent Koszul triality and Guan-Lazarev's work on non-conilpotent Koszul duality. As an application, we use the injective Guan-Lazarev model structure to show that the tensor product is a Quillen bifunctor with respect to these model structures of the second kind.