An injective Model for Twisted Derived Categories and Curved Koszul Triality

TL;DR

This paper introduces a new injective model structure for curved differential graded modules over a curved algebra, establishing a Quillen equivalence with contramodules and connecting different Koszul duality theories.

Contribution

It defines the injective Guan-Lazarev model structure and proves its Quillen equivalence to contramodules over the extended bar construction, bridging two Koszul duality frameworks.

Findings

The model structure is Quillen equivalent to contramodules over the extended bar.

Tensor product acts as a Quillen bifunctor in this setting.

The work connects conilpotent and non-conilpotent Koszul duality theories.

Abstract

Given a 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.