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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2401.07021 |
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| _version_ | 1866914070871932928 |
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| author | Positselski, Leonid Stovicek, Jan |
| author_facet | Positselski, Leonid Stovicek, Jan |
| contents | 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 $\mathsf D^{\mathsf{abs}}(\mathbf{mod}{-}B^\bullet)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_07021 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Contraderived categories of CDG-modules Positselski, Leonid Stovicek, Jan Rings and Algebras Algebraic Topology Category Theory 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 $\mathsf D^{\mathsf{abs}}(\mathbf{mod}{-}B^\bullet)$. |
| title | Contraderived categories of CDG-modules |
| topic | Rings and Algebras Algebraic Topology Category Theory |
| url | https://arxiv.org/abs/2401.07021 |