Filtered derived categories of curved deformations
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866929376315047936 |
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| author | Lehmann, Alessandro Lowen, Wendy |
| author_facet | Lehmann, Alessandro Lowen, Wendy |
| contents | We propose a solution to the "curvature problem" from arXiv:1505.03698 and arXiv:0905.3845 for infinitesimal deformations. Let $k$ be a field, $A$ a dg algebra over $k$ and $A_n = A[t]/(t^{n+1})$ a cdg algebra over $R_n = k[t]/(t^{n+1})$, $n \geq 0$, with reduction $A_n/tA_n = A$. We define the $n$-derived category $D^n(A_n)$ as the quotient of the homotopy category by the modules for which all quotients appearing in the associated graded object are acyclic. We prove this to be a compactly generated triangulated category with a semiorthogonal decomposition by $n + 1$ copies of $D(A)$, in which Positselski's semiderived category embeds admissibly. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_08660 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Filtered derived categories of curved deformations Lehmann, Alessandro Lowen, Wendy K-Theory and Homology 18G80 (Primary), 16E45, 13D10, 18G25 (Secondary) We propose a solution to the "curvature problem" from arXiv:1505.03698 and arXiv:0905.3845 for infinitesimal deformations. Let $k$ be a field, $A$ a dg algebra over $k$ and $A_n = A[t]/(t^{n+1})$ a cdg algebra over $R_n = k[t]/(t^{n+1})$, $n \geq 0$, with reduction $A_n/tA_n = A$. We define the $n$-derived category $D^n(A_n)$ as the quotient of the homotopy category by the modules for which all quotients appearing in the associated graded object are acyclic. We prove this to be a compactly generated triangulated category with a semiorthogonal decomposition by $n + 1$ copies of $D(A)$, in which Positselski's semiderived category embeds admissibly. |
| title | Filtered derived categories of curved deformations |
| topic | K-Theory and Homology 18G80 (Primary), 16E45, 13D10, 18G25 (Secondary) |
| url | https://arxiv.org/abs/2402.08660 |