Contraderived categories of CDG-modules
| Autor: | Positselski, Leonid, Stovicek, Jan |
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| Rok vydání: | 2024 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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 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 $\mathsf D^{\mathsf{abs}}(\mathbf{mod}{-}B^\bullet)$. Comment: LaTeX 2e with xy-pic and one mathb symbol; 68 pages, 7 commutative diagrams; v.2: former Section 5.4 deleted (and replaced by new Remark 5.6) as no longer necessary due to an improvement in Theorem 2.9, new Remarks 4.8 and 6.11 inserted, the final Section 6.9 expanded with Corollary 6.17 improved and new Remark 6.18 added |
| Databáze: | arXiv |
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