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 [1].

This is an expanded note prepared for a STAGE talk at MIT, Fall 2013. Our main references are [1] and [2]. I would like to thank George Boxer for helpful discussions.

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Recall that a *perfectoid field* is a complete nonarchimedean field of residue field of characteristic such that

- The valuation is nondiscrete of rank 1.
- is surjective.

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 [3] for many fields.

Theorem 1
Let be a perfectoid field. The the tilting functor induces an equivalence between of the category of finite extensions of and the category of finite extensions of .

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.

Example 1

- satisfies the condition b) but not a) in the definition of perfectoid fields. If one has the gut to compute the tilt of anyway, one easily finds that .
- satisfies the condition a) but not b) and one finds that too.

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.

Example 2
Let be a perfectoid field. Assume , then is a degree 2 extension of . The key observation is that though is certainly ramified, it is *almost unramified*. Consider the extension at the finite level and . Then by Bezout, is generated over by . One computes the relative different , whose (additive) valuation tends to 0 when . Namely, the ramification gets arbitrarily small when gets arbitrarily large! Another way to say this is that the module of Kahler differentials is killed by for arbitrarily large , i.e., killed by arbitrarily small power of (= killed by the maximal ideal of ).

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 .

Definition 1
Let be a -module. We say is *almost zero* if . We say is *almost zero* if every element of is almost zero, i.e., .

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.

Definition 2
Let be an abelian category. A *Serre subcategory* is a full subcategory of such that for any exact sequence in , one has if and only if . Suppose is a Serre subcategory, the one can form the quotient category , whose objects are the objects of , and for , The quotient category is again an abelian category. By construction, one has a canonical localization functor . If , then in . The quotient category enjoys the following universal property: suppose is another abelian category and is an exact functor such that for any , then uniquely facts through .

Proposition 1
The full subcategory of consisting of almost zero -modules is a Serre subcategory of . Denote the quotient category by .

Proof
Suppose is -torsion, then clearly any sub or quotient of is also -torsion. Conversely, if is an extension of by , where and are -torsion, then kills . But the valuation on is nondiscrete, we have .
¡õ

Definition 3
The quotient category is called the *category of almost -modules*, or -modules. For , we denote to be its image under the localizing functor, i.e., the same object viewed in the almost category .

Remark 1
The category has all formal properties of the category of modules over a commutative ring. So one can define the notion of *almost -algebras*, or *-algebras*: these are *commutative unitary monoid* objects in . Let be a -algebra, one can also define the notation of -modules and -algebras. I should stop boring you by defining them and refer you to the details in [2, 2.2.5]. Just to mention that, for example, an -algebra is an object together with an almost morphism .

Remark 2
One can show that homomorphisms between two almost modules can be described alternatively by This is a -module (an honest module!) with no almost zero elements. We define

Remark 3
There is a right adjoint functor to the localization functor, called *the functor of almost elements* One easily checks (by the previous remark) that is an adjoint pair and

Definition 5
Let be a -algebra and be an -module.

- We say is
*flat*if the functor is flat on -modules. - We say is
*almost projective*if the functor is exact on -modules. (There is the usual notion of projectivity, but it turns out to be ill-behaved: even is not projective over itself.) - Let be a -algebra and be an -module such that and . We say is
*almost finitely generated*if for all , there is some finitely generated -module together with a map such that and are killed by . We further say that is*uniformly finitely generated*if there exists some integer , such that can be generated by elements for all . These notions do not depend on the choice of and . - We similarly define the notion of
*almost finitely presented*modules.

Example 3
Consider again and (). Notice is not finitely generated as a -module: one needs the generators for arbitrarily large . But is an almost finitely generated as a -module. Indeed, for any , the inclusion has cokernel which is killed by . We also see that is almost finitely presented and uniformly finitely generated () as a -module.

Finally, we can define the notion of almost finite etale algebras.

Definition 6
Let be a -algebra and be an -algebra.

- We say is
*unramified*if there is some element such that , and , where and is the multiplication morphism. - We say is
*etale*if it is unramified and is flat as an -module. - We say is
*finite etale*if it is etale and is almost finitely presented as an -module. Denote by the category of finite etale -algebras.

Remark 4
Recall that in commutative algebra, a morphism of finite type is unramified if and only if the diagonal morphism is an open and closed immersion. So one can think of as functions vanishing on the diagonal and as the characteristic function on the diagonal.

Remark 5
Saying is unramified is equivalent to saying that is almost projective as a -module under ; also equivalent to saying that vanishes in the almost category.

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:

Theorem 2
Let be an -algebra. Suppose is flat as an -module and is -adically complete. Then the reduction functor is an equivalence of between the categories and .

The proof of this theorem is not so difficult (an almost version of Nakayama lemma applied to in the almost category). Taking or gives the desired equivalences (2) and (3).
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

Theorem 3 (Almost purity)
Let be a perfectoid field. If is a finite etale algebra, then is finite etale.

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 [4]. Scholze's theory of perfectoid spaces gives a new proof of this hard theorem.

[1]Perfectoid spaces, ArXiv e-prints (2011).

[2]Almost ring theory - sixth release, ArXiv Mathematics e-prints (2002).

[3]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.

[4]Almost étale extensions, Astérisque (2002), no.279, 185--270.