High Frobenius pushforwards generate the bounded derived category
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arXiv
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| Hauptverfasser: | , , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866908791676600320 |
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| author | Ballard, Matthew R. Iyengar, Srikanth B. Lank, Pat Mukhopadhyay, Alapan Pollitz, Josh |
| author_facet | Ballard, Matthew R. Iyengar, Srikanth B. Lank, Pat Mukhopadhyay, Alapan Pollitz, Josh |
| contents | This work concerns generators for the bounded derived category of coherent sheaves over a noetherian scheme $X$ of prime characteristic. The main result is that when the Frobenius map on $X$ is finite, for any compact generator $G$ of $\mathsf{D}(X)$ the Frobenius pushforward $F ^e_*G$ generates the bounded derived category whenever $p^e$ is larger than the codepth of $X$, an invariant that is a measure of the singularity of $X$. The conclusion holds for all positive integers $e$ when $X$ is locally complete intersection. The question of when one can take $G=\mathcal{O}_X$ is also investigated. For smooth projective complete intersections it reduces to a question of generation of the Kuznetsov component. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_18085 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | High Frobenius pushforwards generate the bounded derived category Ballard, Matthew R. Iyengar, Srikanth B. Lank, Pat Mukhopadhyay, Alapan Pollitz, Josh Algebraic Geometry Commutative Algebra 14A30 (primary), 13A35, 14G17, 13D09 This work concerns generators for the bounded derived category of coherent sheaves over a noetherian scheme $X$ of prime characteristic. The main result is that when the Frobenius map on $X$ is finite, for any compact generator $G$ of $\mathsf{D}(X)$ the Frobenius pushforward $F ^e_*G$ generates the bounded derived category whenever $p^e$ is larger than the codepth of $X$, an invariant that is a measure of the singularity of $X$. The conclusion holds for all positive integers $e$ when $X$ is locally complete intersection. The question of when one can take $G=\mathcal{O}_X$ is also investigated. For smooth projective complete intersections it reduces to a question of generation of the Kuznetsov component. |
| title | High Frobenius pushforwards generate the bounded derived category |
| topic | Algebraic Geometry Commutative Algebra 14A30 (primary), 13A35, 14G17, 13D09 |
| url | https://arxiv.org/abs/2303.18085 |