---
tag: K768V9ZS7R
kind: definition
name: Perfectoid space
msc: [14G45, 14G22, 12J25]
defines: Q9Y4CJE1X5
uses: [1H5F1BP6XS, W168DS966V, 0T3WVTD675, 06GZ956657, YAH6MZVG7K, BT6T702C28]
sources:
  - url: https://en.wikipedia.org/wiki/Perfectoid_space
---

## Definition

Let $p$ be a prime.
Let $K$ be a field, complete for a [[non-archimedean absolute value|1H5F1BP6XS]] $\abs{\cdot}$, and let $\mathcal{O}_K = \set{x : \abs{x} \le 1}$.
The field $K$ is a *perfectoid field* if:

1. The value group $\abs{K^\times}$ is not discrete.
2. The residue field of $\mathcal{O}_K$ has characteristic $p$.
3. The [[Frobenius map|0T3WVTD675]] $x \mapsto x^p$ on $\mathcal{O}_K / p\mathcal{O}_K$ is surjective.


Let $K$ be a perfectoid field.
A *perfectoid $K$-algebra* is a Banach $K$-algebra $R$ such that the subring $R^\circ$ of power-bounded elements is bounded and the Frobenius map $R^\circ / p R^\circ \to R^\circ / p R^\circ$ is surjective.
A *perfectoid affinoid $K$-algebra* is a pair $(R, R^+)$ with $R$ a perfectoid $K$-algebra and $R^+ \subseteq R^\circ$ an open and integrally closed subring.
A *perfectoid space* over $K$ is an adic space over $K$ that is covered by open subspaces isomorphic to $\operatorname{Spa}(R, R^+)$ with $(R, R^+)$ perfectoid affinoid.

## Remarks

Example: the [[completion|W168DS966V]] $K$ of $\Q_p(p^{1/p^\infty}) = \bigcup_n \Q_p(p^{1/p^n})$ is a perfectoid field.
Its value group is $p^{\Z[1/p]}$, which is not discrete, and every element of $\mathcal{O}_K / p$ is a polynomial in the $p^{1/p^n}$, each of which is the $p$-th power of $p^{1/p^{n+1}}$.
The completed algebraic closure $\C_p$ of the [[$p$-adic numbers|BT6T702C28]] is also perfectoid.

Non-example: $\Q_p$ is not perfectoid.
The Frobenius map on $\Z_p / p = \F_p$ is surjective, but the value group $p^{\Z}$ is discrete, so the first condition fails.

The tilt of $K$ is $K^\flat$, the fraction field of $\varprojlim_{x \mapsto x^p} \mathcal{O}_K / p$; it is a perfectoid field of characteristic $p$.
Scholze's tilting equivalence identifies perfectoid spaces over $K$ with perfectoid spaces over $K^\flat$.