For any perfectoid field K there is a tilt K^♭, which is a perfectoid field of finite characteristic p. As a set, it may be defined as

${\displaystyle K^{\flat }=\varprojlim _{x\mapsto x^{p}}K.}$ 

Explicitly, an element of K^♭ is an infinite sequence (x₀, x₁, x₂, ...) of elements of K such that x_i = x^p
_i+1. The multiplication in K^♭ is defined termwise, while the addition is more complicated. If K has finite characteristic, then K ≅ K^♭. If K is the p-adic completion of ${\displaystyle \mathbb {Q} _{p}(p^{1/p^{\infty }})}$ , then K^♭ is the t-adic completion of ${\displaystyle \mathbb {F} _{p}((t))(t^{1/p^{\infty }})}$ .

There are notions of perfectoid algebras and perfectoid spaces over a perfectoid field K, roughly analogous to commutative algebras and schemes over a field. The tilting operation extends to these objects. If X is a perfectoid space over a perfectoid field K, then one may form a perfectoid space X^♭ over K^♭. The tilting equivalence is a theorem that the tilting functor (-)^♭ induces an equivalence of categories between perfectoid spaces over K and perfectoid spaces over K^♭. Note that while a perfectoid field of finite characteristic may have several non-isomorphic "untilts", the categories of perfectoid spaces over them would all be equivalent.

Almost purity theorem edit

This equivalence of categories respects some additional properties of morphisms. Many properties of morphisms of schemes have analogues for morphisms of adic spaces. The almost purity theorem for perfectoid spaces is concerned with finite étale morphisms. It's a generalization of Faltings's almost purity theorem in p-adic Hodge theory. The name is alluding to almost mathematics, which is used in a proof, and a distantly related classical theorem on purity of the branch locus.^[2]

The statement has two parts. Let K be a perfectoid field.

• If X → Y is a finite étale morphism of adic spaces over K and Y is perfectoid, then X also is perfectoid;
• A morphism X → Y of perfectoid spaces over K is finite étale if and only if the tilt X^♭ → Y^♭ is finite étale over K^♭.

Since finite étale maps into a field are exactly finite separable field extensions, the almost purity theorem implies that for any perfectoid field K the absolute Galois groups of K and K^♭ are isomorphic.

• Perfect field

• Bhatt, Bhargav. "What is a ... Perfectoid Space?" (PDF). Bulletin of the AMS. Retrieved 2 January 2020.
• "What are "perfectoid spaces"?". MathOverflow.
• Foundations of Perfectoid Spaces by Matthew Morrow
• Lean perfectoid spaces. The definition of perfectoid spaces formalized in the Lean theorem prover