Beyond the Fontaine-Wintenberger theorem
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866916650434953216 |
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| author | Jahnke, Franziska Kartas, Konstantinos |
| author_facet | Jahnke, Franziska Kartas, Konstantinos |
| contents | Given a perfectoid field, we find an elementary extension and a henselian defectless valuation on it, whose value group is divisible and whose residue field is an elementary extension of the tilt. This specializes to the almost purity theorem over perfectoid valuation rings and Fontaine-Wintenberger. Along the way, we prove an Ax-Kochen/Ershov principle for certain deeply ramified fields, which also uncovers some new model-theoretic phenomena in positive characteristic. Notably, we get that the perfect hull of $\mathbb{F}_p(t)^h$ is an elementary substructure of the perfect hull of $\mathbb{F}_p(\!(t)\!)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_05881 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Beyond the Fontaine-Wintenberger theorem Jahnke, Franziska Kartas, Konstantinos Commutative Algebra Logic Given a perfectoid field, we find an elementary extension and a henselian defectless valuation on it, whose value group is divisible and whose residue field is an elementary extension of the tilt. This specializes to the almost purity theorem over perfectoid valuation rings and Fontaine-Wintenberger. Along the way, we prove an Ax-Kochen/Ershov principle for certain deeply ramified fields, which also uncovers some new model-theoretic phenomena in positive characteristic. Notably, we get that the perfect hull of $\mathbb{F}_p(t)^h$ is an elementary substructure of the perfect hull of $\mathbb{F}_p(\!(t)\!)$. |
| title | Beyond the Fontaine-Wintenberger theorem |
| topic | Commutative Algebra Logic |
| url | https://arxiv.org/abs/2304.05881 |