Contraderived categories of CDG-modules

For any CDG-ring $B^{\bullet }=(B^{\ast },d,h)$, 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) ${\mathsf{D}}^{\mathsf{b}\mathsf{c}\mathsf{t}\mathsf{r}}(B^{\bullet }-\mathbf{M}\mathbf{o}\mathbf{d})$ 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 $B^{\ast }$ to be graded right coherent, we show that the contraderived category ${\mathsf{D}}^{\mathsf{b}\mathsf{c}\mathsf{t}\mathsf{r}}(B^{\bullet }-\mathbf{M}\mathbf{o}\mathbf{d})$ 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 ${\mathsf{D}}^{\mathsf{b}\mathsf{c}\mathsf{o}}(\mathbf{M}\mathbf{o}\mathbf{d}-B^{\bullet })$. Specifically, the latter triangulated category is the idempotent completion of the absolute derived category of finitely presented right CDG-modules ${\mathsf{D}}^{\mathsf{a}\mathsf{b}\mathsf{s}}(\mathbf{m}\mathbf{o}\mathbf{d}-B^{\bullet })$.