In mathematics, perfectoid spaces are adic spaces of special kind, which occur in the study of problems of "mixed characteristic", such as local fields of characteristic zero which have residue fields of characteristic prime p. A perfectoid field is a complete topological field K whose topology is induced by a nondiscrete valuation of rank 1, such that the Frobenius endomorphism Φ is surjective on K°/p where K° denotes the ring of power-bounded elements.
Technical tools for making this precise are the tilting equivalence and the almost purity theorem.
[1] 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 Explicitly, an element of K♭ is an infinite sequence (x0, x1, x2, ...) of elements of K such that xi = xpi+1.
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♭.
This equivalence of categories respects some additional properties of morphisms.
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.