nLab perfectoid field

Definition

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

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

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

where $\Phi$ is the Frobenius map.

Related entries

• perfectoid space