Contraderived categories of CDG-modules

Abstract

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

LaTeX 2e with xy-pic and one mathb symbol; 81 pages, 8 commutative diagrams; v.2: former Section 5.4 replaced by Remark 5.6, new Remarks 4.8 and 6.11 inserted, the final Section 6.9 expanded; v.3: Introduction rearranged and polished; former Section 5.1 is now Section 4.12; new Sections 4.14 and 6.10, with examples, added; motivating explanations added in Section 5; v.4: small things corrected