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arXiv · 2210.08567

A criterion for perfectoid fields

Ehsan Shahoseini·Kiran S. Kedlaya
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Abstract

The tilting correspondence is a fundamental property of perfectoid fields. In this note, we show that the tilting construction can also be used to detect perfectoid fields among nonarchimedean fields. In particular, for $K$ a complete subfield of $\mathbb{C}_p$ (a completed algebraic closure of $\mathbb{Q}_p$), $K$ is perfectoid if and only if its tilt is not algebraic over $\mathbb{F}_p$. We also include some conjectures on APF (arithmetically profinite) extensions, perfectoid fields, and their relations.

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Ehsan Shahoseini, Kiran S. Kedlaya. 2022-10-16. A criterion for perfectoid fields. https://arxiv.org/abs/2210.08567

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