Generators vs. classical generators in derived categories of curves
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908619141808128 |
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| author | Pirozhkov, Dmitrii |
| author_facet | Pirozhkov, Dmitrii |
| contents | This is mostly an expository note about an example communicated to the author by Aise Johan de Jong. In a triangulated category $T$ an object $G$ is said to be a classical generator when the smallest triangulated subcategory containing $G$ coincides with the whole $T$, and it is said to be a generator when the orthogonal complement to $G$ in $T$ is zero, i.e., when any non-zero object of $T$ admits a non-zero map from a shift of $G$. Any classical generator is a generator, but not vice versa. We discuss a simple algebro-geometric example of a non-classical generator in the derived category of coherent sheaves on any smooth proper curve of genus $g \geq 2$. We also overview what is known and what is not known, in general, about generators and classical generators on curves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_25558 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generators vs. classical generators in derived categories of curves Pirozhkov, Dmitrii Algebraic Geometry This is mostly an expository note about an example communicated to the author by Aise Johan de Jong. In a triangulated category $T$ an object $G$ is said to be a classical generator when the smallest triangulated subcategory containing $G$ coincides with the whole $T$, and it is said to be a generator when the orthogonal complement to $G$ in $T$ is zero, i.e., when any non-zero object of $T$ admits a non-zero map from a shift of $G$. Any classical generator is a generator, but not vice versa. We discuss a simple algebro-geometric example of a non-classical generator in the derived category of coherent sheaves on any smooth proper curve of genus $g \geq 2$. We also overview what is known and what is not known, in general, about generators and classical generators on curves. |
| title | Generators vs. classical generators in derived categories of curves |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2510.25558 |