Perfectoid space

Builds on Non-Archimedean absolute value Completion of a valued field Frobenius endomorphism Banach space Rigid analytic space p-adic number

Definition

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

The value group $\abs{K^\times}$ is not discrete.
The residue field of $\mathcal{O}_K$ has characteristic $p$ .
The Frobenius map $x \mapsto x^p$ 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 $K$ 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 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 $K^\flat$ .

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Tag	`K768V9ZS7R`
URI	`m2://K768V9ZS7R`
Kind	definition
Writing	W0 stub
MSC	14G45, 14G22, 12J25
Defines	Perfectoid space
Uses	Non-Archimedean absolute value, Completion of a valued field, Frobenius endomorphism, Banach space, Rigid analytic space, p-adic number
Changed	2 November 2026

Perfectoid space

Definition

Remarks

Reading path

Further reading

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