A criterion for perfectoid fields

2022/10/16 by Shahoseini, Ehsan, Kedlaya, Kiran S.
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2210.08567

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 ℂ_p (a completed algebraic closure of ℚ_p), K is perfectoid if and only if its tilt is not algebraic over \mathbbF_p. We also include some conjectures on APF (arithmetically profinite) extensions, perfectoid fields, and their relations.

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