Coherent six-functor formalisms: Pro vs Solid
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866910235433631744 |
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| author | Ren, Fei |
| author_facet | Ren, Fei |
| contents | In the classical theory for coherent sheaves, the only missing piece in the Grothendieck six-functor formalism picture is $j_!$ for an open immersion $j$. Towards fixing this gap, Deligne proposed a construction of $j_!$ by extending the sheaf class to pro sheaves, while Clausen-Scholze provided another solution by extending the sheaf class to solid modules. In this work, we prove that Deligne's construction coincides with the Clausen-Scholze construction via a natural functor, whose restriction to the full subcategory of Mittag-Leffler pro-systems is fully faithful. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_21082 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Coherent six-functor formalisms: Pro vs Solid Ren, Fei Algebraic Geometry General Topology In the classical theory for coherent sheaves, the only missing piece in the Grothendieck six-functor formalism picture is $j_!$ for an open immersion $j$. Towards fixing this gap, Deligne proposed a construction of $j_!$ by extending the sheaf class to pro sheaves, while Clausen-Scholze provided another solution by extending the sheaf class to solid modules. In this work, we prove that Deligne's construction coincides with the Clausen-Scholze construction via a natural functor, whose restriction to the full subcategory of Mittag-Leffler pro-systems is fully faithful. |
| title | Coherent six-functor formalisms: Pro vs Solid |
| topic | Algebraic Geometry General Topology |
| url | https://arxiv.org/abs/2506.21082 |