nLab
perfectoid field

Definition

A perfectoid field KK is a complete non-archimedean field KK of residue characteristic pp, equipped with a non-discrete valuation of rank 1, such that the Frobenius map Φ:𝒪 K/p𝒪 K/p\Phi: \mathcal{O}_K/p \to \mathcal{O}_K/p is surjective, where 𝒪 KK\mathcal{O}_K \subset K is the subring of elements of norm 1\leq 1.

Given a perfectoid field, KK, one can form a second perfectoid field K K^{\flat}, always of characteristic pp, given as the fraction field of

𝒪 K =lim Φ𝒪 K/p, \mathcal{O}_{K^{\flat}} = \underset{\leftarrow}{lim}_{\Phi} \mathcal{O}_K/p,

where Φ\Phi is the Frobenius map.

Last revised on July 7, 2017 at 15:41:45. See the history of this page for a list of all contributions to it.