An injective Model for Twisted Derived Categories and Curved Koszul Triality

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.

💡 Research Summary

The paper introduces a new model structure on the category of curved differential graded (CDG) modules over a curved differential graded algebra (A). Building on the work of Guan‑Lazarev, who defined a “projective” model structure (denoted (A\text{-mod}{II}^{\text{proj}})) whose weak equivalences are detected by Hom‑functors from finitely generated twisted modules, the authors construct a dual “injective” model structure, denoted (A\text{-mod}{II}^{\text{inj}}).

The key technical device is the extended bar construction (\widehat B A), a possibly non‑conilpotent CDG coalgebra associated to (A). There is a natural Maurer–Cartan element (\tau:\widehat B A\to A) which yields an adjunction
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