Contraderived categories of CDG-modules
Leonid Positselski, Jan Stovicek
TL;DR
The paper extends Becker’s contraderived framework to curved DG-rings by analyzing CDG-modules and establishes a pair of central equivalences: the projective contraderived category is equivalent to the flat contraderived category, and the flat contraderived category is equivalent to the flat-cotorsion (coderived-like) category. It proves cotorsion and graded-cotorsion coincide for CDG-modules and shows this structure yields a stable, Quillen-compatible homotopy theory with abelian-model realizations. Under graded-right coherence of the underlying graded ring, the contraderived category is compactly generated, and its compact objects are described via the absolute derived category of finitely presented CDG-modules, anti-equivalently related to the coderived category’s compact objects. The results encompass matrix factorizations and de Rham-type CDG-structures, providing a broad, robust framework for second-kind derived categories in nontrivial curved settings. Overall, the work unifies several perspectives on homotopy categories of CDG-modules and delivers concrete generators and dualities that have potential applications in geometry and mathematical physics.
Abstract
For any CDG-ring $B^\bullet=(B^*,d,h)$, we show that the homotopy category of graded-projective (left) CDG-modules over $B^\bullet$ is equivalent to the quotient category of the homotopy category of graded-flat CDG-modules by its full triangulated subcategory of flat CDG-modules. The contraderived category (in the sense of Becker) $\mathsf D^{\mathsf{bctr}}(B^\bullet{-}\mathbf{Mod})$ is the common name for these two triangulated categories. We also prove that the classes of cotorsion and graded-cotorsion CDG-modules coincide, and the contraderived category of CDG-modules is equivalent to the homotopy category of graded-flat graded-cotorsion CDG-modules. Assuming the graded ring $B^*$ to be graded right coherent, we show that the contraderived category $\mathsf D^{\mathsf{bctr}}(B^\bullet{-}\mathbf{Mod})$ is compactly generated and its full subcategory of compact objects is anti-equivalent to the full subcategory of compact objects in the coderived category of right CDG-modules $\mathsf D^{\mathsf{bco}}(\mathbf{Mod}{-}B^\bullet)$. Specifically, the latter triangulated category is the idempotent completion of the absolute derived category of finitely presented right CDG-modules $\mathsf D^{\mathsf{abs}}(\mathbf{mod}{-}B^\bullet)$.