Contraderived categories of CDG-modules

For any CDG-ring

, we show that the homotopy category of graded-projective (left) CDG-modules over

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)

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

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

. Specifically, the latter triangulated category is the idempotent completion of the absolute derived category of finitely presented right CDG-modules

.