Prismatic cohomology relative to $δ$-rings
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866915987729678336 |
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| author | Antieau, Benjamin Krause, Achim Nikolaus, Thomas |
| author_facet | Antieau, Benjamin Krause, Achim Nikolaus, Thomas |
| contents | We develop prismatic and syntomic cohomology relative to a $δ$-ring. This simultaneously generalizes Bhatt and Scholze's absolute and relative prismatic cohomology and shows that the latter, which was defined relative to a prism, is in fact independent of the prism structure and only depends on the underlying $δ$-ring. We give several possible definitions of our new version of prismatic cohomology: a site theoretic definition, one using prismatic crystals, and a stack theoretic definition. These are equivalent under mild syntomicity hypotheses. As an application, we note how the theory of prismatic cohomology of filtered rings arises naturally in this context. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_12770 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Prismatic cohomology relative to $δ$-rings Antieau, Benjamin Krause, Achim Nikolaus, Thomas Algebraic Geometry K-Theory and Homology 19D55, 14F30 We develop prismatic and syntomic cohomology relative to a $δ$-ring. This simultaneously generalizes Bhatt and Scholze's absolute and relative prismatic cohomology and shows that the latter, which was defined relative to a prism, is in fact independent of the prism structure and only depends on the underlying $δ$-ring. We give several possible definitions of our new version of prismatic cohomology: a site theoretic definition, one using prismatic crystals, and a stack theoretic definition. These are equivalent under mild syntomicity hypotheses. As an application, we note how the theory of prismatic cohomology of filtered rings arises naturally in this context. |
| title | Prismatic cohomology relative to $δ$-rings |
| topic | Algebraic Geometry K-Theory and Homology 19D55, 14F30 |
| url | https://arxiv.org/abs/2310.12770 |