the tilting equivalences!

Integers moving,

Numbers evolving

Integers

The tilting equivalence is a theorem that the tilting functor (-)♭ induces an equivalence of categories perfectoid spaces over Kand perfectoid spaces over K♭.

Note that while a perfectoid field of the finite characteristic may have several non-isomorphic "untilts," the categories of perfectoid spaces over them would all be equivalent.

The almost purity theorem for perfectoid spaces concerns finite étale morphisms.

It's a generalization of Faltings's almost purity theorem in p-adic Hodge theory.

The name alludes to almost mathematics, which is used in a proof, and a distantly related classical theorem on the purity of the branch locus.