Contraderived categories of CDG-modules

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 $\mathsf D^{\mathsf{abs}}(\mathbf{mod}{-}B^\bullet)$.