The tilting functor induces an equivalence between the category of finite extensions of a perfectoid field and the category of finite extensions of its tilt . The proof of this fact (and further generalization in families) relies on Faltings' almost mathematics. We shall introduce the necessary background in almost ring theory and sketch the proof following Chapter 4 of Scholze's paper .
Recall that a perfectoid field is a complete nonarchimedean field of residue field of characteristic such that
We also defined a tilting operation: let such that . Then where is an element of such that . We have
The following is the main theorem of today's talk, which already appeared in the classical work of Fontaine-Wintenberger  for many fields.
In particular, one obtains a canonical isomorphism between the absolute Galois group of (a characteristic 0 field) and the absolute Galois group of (a characteristic field) . To convince you that this is really a miracle, let us consider the following two (non-)examples.
In both cases, one loses tons of information when passing to the tilt and the above main theorem is far from true!
So how does one prove such a miraculous theorem? The main tool is Faltings' almost mathematics. Let us look at the following simple example to motivate this idea.
So we would like to consider a category of -modules where the -torsion are systematically ignored ( almost -modules, or -modules) and use this category to define the notion of almost finite etale -algebra rigorously. In view of the above example, a finite (= finite etale) extension is expected to extend automatically to an almost finite etale extension on the integral level. This is known as almost purity (analogous to the Nagata-Zariski's purity theorem on the branch locus in algebraic geometry) and lies in the core of the proof of the above stated main theorem.
Fix a perfectoid field with valuation ring and maximal ideal .
We would like to "quotient out" all almost zero modules. We recall the machinery to do so: the quotient of an abelian category by a Serre subcategory.
Finally, we can define the notion of almost finite etale algebras.
The following diagram illustrates the structure of the proof of the main theorem. We would like to show that all the arrows in the diagram are equivalences. Here the left arrow (1) is the generic fiber functor and the right arrow (2) is the reduction functor . The arrows (3) and (4) are similarly defined.
Step 1 Show (2) and (3) are equivalences. This boils down to prove that any finite etale algebras lift uniquely over nilpotents. This is not so surprising and is true in much more generality:
Step 2 Show (1) and (4) are fully faithful. This can be checked directly and is also a consequence of the (easy) equivalence between the larger categories of perfectoid algebras over and . In particular, the essential image of (1) consists of finite etale algebras such that is finite etale. To show the equivalence (1) and (4), it suffices to show that
Step 3 Show the almost purity for perfectoid fields of characteristic (hence (4) is an equivalence). This is not difficult by the existence of Frobenius. Here is the outline of the argument. Suppose is the idempotent from the finite etale algebra , then there exists some integer such that . Since the Frobenius is surjective, as well for any . Since can be arbitrarily large, we obtain an almost idempotent for as desired.
Step 4 Show the almost purity for perfectoid fields of characteristic 0 (hence (1) is an equivalence). This is very difficult. One can find a proof in [2, 6.6.6] using ramification theory (alluded in the beginning of this talk) . Over more general base, the almost purity is proved by Faltings in . Scholze's theory of perfectoid spaces gives a new proof of this hard theorem.
Perfectoid spaces, ArXiv e-prints (2011).
Almost ring theory - sixth release, ArXiv Mathematics e-prints (2002).
Le ``corps des normes'' de certaines extensions algébriques de corps locaux, C. R. Acad. Sci. Paris Sér. A-B 288 (1979), no.6, A367--A370.
Almost étale extensions, Astérisque (2002), no.279, 185--270.