The tilting functor induces an equivalence between the category of finite extensions of a perfectoid field $K$ and the category of finite extensions of its tilt $K^\flat$. 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.

TopMotivation

Recall that a perfectoid field $K$ is a complete nonarchimedean field of residue field of characteristic $p>0$ such that

  1. The valuation is nondiscrete of rank 1.
  2. $\Phi: K^\circ/p\rightarrow K^\circ/p,\quad x\mapsto x^p$ is surjective.

We also defined a tilting operation: let $\varpi\in K$ such that $|p|\le|\varpi|<1$. Then $$K^{\flat}=(\varprojlim_{\Phi} K^\circ/\varpi)[(\varpi^{\flat})^{-1}],$$ where $\varpi^{\flat}$ is an element of $\varprojlim_{\Phi} K^\circ/\varpi$ such that $|(\varpi^{\flat})^\sharp|=|\varpi|$. We have $$K^\circ/\varpi\cong K^{\flat \circ}/\varpi^{\flat}.$$

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 $K$ be a perfectoid field. The the tilting functor $L\mapsto L^{\flat}$ induces an equivalence between of the category of finite extensions of $K$ and the category of finite extensions of $K^{\flat}$.

In particular, one obtains a canonical isomorphism between the absolute Galois group of (a characteristic 0 field) $K$ and the absolute Galois group of (a characteristic $p$ field) $K^{\flat}$. To convince you that this is really a miracle, let us consider the following two (non-)examples.

Example 1
  • $K=\mathbb{Q}_p$ satisfies the condition b) but not a) in the definition of perfectoid fields. If one has the gut to compute the tilt of $K$ anyway, one easily finds that $K^{\flat}=\mathbb{F}_p$.
  • $K=\mathbb{Q}_p(p^{1/\ell^\infty})$ satisfies the condition a) but not b) and one finds that $K^{\flat}=\mathbb{F}_p$ 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 $K=\mathbb{Q}_p(p^{1/p^\infty})^{\wedge}$ be a perfectoid field. Assume $p\ne2$, then $L=K(p^{1/2})$ is a degree 2 extension of $K$. The key observation is that though $L/K$ is certainly ramified, it is almost unramified. Consider the extension at the finite level $K_n=\mathbb{Q}_p(p^{1/p^n})$ and $L_n=K_n(p^{1/2})$. Then by Bezout, $L_n^\circ$ is generated over $K_n^\circ$ by $p^{1/2p^n}$. One computes the relative different $\mathcal{D}_{L_n^\circ/K_n^\circ}=(p^{1/2p^n})$, whose (additive) valuation tends to 0 when $n\rightarrow\infty$. Namely, the ramification gets arbitrarily small when $n$ gets arbitrarily large! Another way to say this is that the module of Kahler differentials $\Omega_{L_n^\circ/K_n^\circ}$ is killed by $p^{1/p^n}$ for arbitrarily large $n$, i.e., killed by arbitrarily small power of $p$ (= killed by the maximal ideal $\mathfrak{m}$ of $K^\circ$).

So we would like to consider a category of $K^\circ$-modules where the $\mathfrak{m}$-torsion are systematically ignored ( almost $K^\circ$-modules, or $K^{\circ a}$-modules) and use this category to define the notion of almost finite etale $K^\circ$-algebra rigorously. 
$$\xymatrix{ K^\circ\Mod \ar[rr]^{\text{ignoring all $p$-power torsion}} \ar[rd]_{\text{ignoring only}\atop\text{small $p$-power torsion}} &  & K\Mod & \text{(generic fiber)}\\ & K^{\circ a}\Mod \ar[ru] & &\text{(slightly generic fiber)} }$$
In view of the above example, a finite (= finite etale) extension $L/K$ is expected to extend automatically to an almost finite etale extension $L^\circ/K^\circ$ 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.

TopAlmost modules and almost algebras

Fix a perfectoid field $K$ with valuation ring $K^\circ$ and maximal ideal $\mathfrak{m}$.

Definition 1 Let $M$ be a $K^\circ$-module. We say $x\in M$ is almost zero if $\mathfrak{m}x=0$. We say $M$ is almost zero if every element of $M$ is almost zero, i.e., $\mathfrak{m} M=0$.

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 $\mathcal{A}$ be an abelian category. A Serre subcategory is a full subcategory $\mathcal{B}$ of $\mathcal{A}$ such that for any exact sequence $$0\rightarrow M'\rightarrow M\rightarrow M''\rightarrow 0$$ in $\mathcal{A}$, one has $M\in \mathcal{B}$ if and only if $M',M''\in \mathcal{B}$. Suppose $\mathcal{B}$ is a Serre subcategory, the one can form the quotient category $\mathcal{A}/\mathcal{B}$, whose objects are the objects of $\mathcal{A}$, and for $M, N\in \mathcal{A}$, $$\Hom_{\mathcal{A/}\mathcal{B}}(M,N)=\varinjlim_{\alpha:M'\hookrightarrow M, \beta: N\twoheadrightarrow N''\atop \ker\alpha,\coker\beta\in\mathcal{B}}\Hom(M', N'').$$ The quotient category is again an abelian category. By construction, one has a canonical localization functor $Q: \mathcal{A}\rightarrow\mathcal{A}/\mathcal{B}$. If $M\in\mathcal{B}$, then $Q(M)\cong0$ in $\mathcal{A}/\mathcal{B}$. The quotient category enjoys the following universal property: suppose $\mathcal{C}$ is another abelian category and $F: \mathcal{A}\rightarrow\mathcal{C}$ is an exact functor such that $F(M)=0$ for any $M\in \mathcal{B}$, then $F$ uniquely facts through $\mathcal{A}/\mathcal{B}$.
Proposition 1 The full subcategory of $K^\circ\Mod$ consisting of almost zero $K^\circ$-modules is a Serre subcategory of $K^\circ\Mod$. Denote the quotient category by $K^{\circ a}\Mod$.
Proof Suppose $M$ is $\mathfrak{m}$-torsion, then clearly any sub or quotient of $M$ is also $\mathfrak{m}$-torsion. Conversely, if $M$ is an extension of $M''$ by $M'$, where $M'$ and $M''$ are $\mathfrak{m}$-torsion, then $\mathfrak{m}^2$ kills $M$. But the valuation on $K$ is nondiscrete, we have $\mathfrak{m}^2=\mathfrak{m}$.
Definition 3 The quotient category $K^{\circ a}\Mod$ is called the category of almost $K^\circ$-modules, or $K^{\circ a}$-modules. For $M\in K^\circ\Mod$, we denote $M^a$ to be its image under the localizing functor, i.e., the same object viewed in the almost category $K^{\circ a}\Mod$.
Definition 4 Define $$M^a \otimes_{K^{\circ a}} N^a=(M \otimes_{K^\circ} N)^a.$$ It is a well-defined on $K^{\circ a}\Mod$ and makes $K^{\circ a }\Mod$ an abelian tensor category.
Remark 1 The category $K^{\circ a}\Mod$ has all formal properties of the category of modules over a commutative ring. So one can define the notion of almost $K^\circ$-algebras, or $K^{\circ a}$-algebras: these are commutative unitary monoid objects in $K^{\circ a}\Mod$. Let $A$ be a $K^{\circ a}$-algebra, one can also define the notation of $A$-modules and $A$-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 $A$-algebra is an object $B\in K^{\circ a}\Mod$ together with an almost morphism $A\rightarrow B$.
Remark 2 One can show that homomorphisms between two almost modules can be described alternatively by $$\Hom_{K^{\circ a}}(M^a,N^a)=\Hom_{K^\circ}(\mathfrak{m} \otimes_{K^\circ} M, N).$$ This is a $K^\circ$-module (an honest module!) with no almost zero elements. We define $$\alHom(M^a,N^a)=\Hom_{K^\circ}(M^a,N^a)^a.$$
Remark 3 There is a right adjoint functor to the localization functor, called the functor of almost elements $$M\mapsto M_*{}=\Hom_{K^{\circ a}}(K^{\circ a}, M).$$ One easily checks (by the previous remark) that $(-^a,- _ {*})$ is an adjoint pair and $$(M_*)^a\cong M,\quad (M^a)_*\cong\Hom_{K^\circ}(\mathfrak{m}, M).$$

TopAlmost properties

Definition 5 Let $A$ be a $K^{\circ a}$-algebra and $M$ be an $A$-module.
  • We say $M$ is flat if the functor $X\mapsto X \otimes_A M$ is flat on $A$-modules.
  • We say $M$ is almost projective if the functor $X\mapsto \alHom_A(M,X)$ is exact on $A$-modules. (There is the usual notion of projectivity, but it turns out to be ill-behaved: even $K^{\circ a}$ is not projective over itself.)
  • Let $R$ be a $K^\circ$-algebra and $N$ be an $R$-module such that $M=N^a$ and $A=R^a$. We say $M$ is almost finitely generated if for all $\varepsilon\in \mathfrak{m}$, there is some finitely generated $R$-module $N_\varepsilon$ together with a map $f_\varepsilon: N_\varepsilon\rightarrow N$ such that $\ker f_\varepsilon$ and $\coker f_\varepsilon$ are killed by $\varepsilon$. We further say that $M$ is uniformly finitely generated if there exists some integer $n$, such that $N_\varepsilon$ can be generated by $n$ elements for all $\varepsilon$. These notions do not depend on the choice of $R$ and $N$.
  • We similarly define the notion of almost finitely presented modules.
Example 3 Consider again $K=\mathbb{Q}_p(p^{1/p^\infty})^\wedge$ and $L=K(p^{1/2})$ ($p\ne2$). Notice $L^\circ$ is not finitely generated as a $K^\circ$-module: one needs the generators $p^{1/2p^n}$ for arbitrarily large $n$. But $L^{\circ a}$ is an almost finitely generated as a $K^{\circ a}$-module. Indeed, for any $\varepsilon=p^{1/p^n}\in \mathfrak{m}$, the inclusion $f_\varepsilon: K^\circ \oplus p^{1/2p^n}K^\circ\rightarrow L^\circ$ has cokernel which is killed by $\varepsilon$. We also see that $L^{\circ a}$ is almost finitely presented and uniformly finitely generated ($n=2$) as a $K^{\circ a}$-module.

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

Definition 6 Let $A$ be a $K^{\circ a}$-algebra and $B$ be an $A$-algebra.
  • We say $A\rightarrow B$ is unramified if there is some element $e\in (B \otimes_A B)_*$ such that $e^2=e$, $\mu(e)=1$ and $I_*e=0$, where $I=\ker(\mu: B \otimes_A B\rightarrow B)$ and $\mu$ is the multiplication morphism.
  • We say $A\rightarrow B$ is etale if it is unramified and $B$ is flat as an $A$-module.
  • We say $A\rightarrow B$ is finite etale if it is etale and $B$ is almost finitely presented as an $A$-module. Denote by $A_{\mathrm{fet}}$ the category of finite etale $A$-algebras.
Remark 4 Recall that in commutative algebra, a morphism $\Spec B\rightarrow\Spec A$ of finite type is unramified if and only if the diagonal morphism $\Spec B\rightarrow \Spec B\times_{\Spec A}\Spec B$ is an open and closed immersion. So one can think of $I$ as functions vanishing on the diagonal and $e$ as the characteristic function on the diagonal.
Remark 5 Saying $A\rightarrow B$ is unramified is equivalent to saying that $B$ is almost projective as a $B \otimes_A B$-module under $\mu$; also equivalent to saying that $\Omega_{B/A}:=I/I^2$ vanishes in the almost category.

TopAlmost a sketch of the proof

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. $$\xymatrix{K_{\mathrm{fet}} & K^{\circ a}_{\mathrm{fet}} \ar[r]^-{(2)} \ar[l]_-{(1)} & (K^{\circ a}/\varpi)_{\mathrm{fet}} \ar@{=}[d] \\ K^{\flat}_{\mathrm{fet}} & K^{\flat \circ a}_{\mathrm{fet}} \ar[r]^-{(3)} \ar[l]_-{(4)} & (K^{\flat \circ a}/\varpi^{\flat})_{\mathrm{fet}}}$$ Here the left arrow (1) is the generic fiber functor $A\mapsto A_*[\varpi^{-1}]$ and the right arrow (2) is the reduction functor $A\mapsto A \otimes_{K^{\circ a}} K^{\circ a}/\varpi$. 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 $A$ be an $K^{\circ a}$-algebra. Suppose $A$ is flat as an $K^{\circ a}$-module and $A$ is $\varpi$-adically complete. Then the reduction functor $B\mapsto B \otimes_A A/\varpi$ is an equivalence of between the categories $A_{\mathrm{fet}}$ and $A/\varpi_{\mathrm{fet}}$.

The proof of this theorem is not so difficult (an almost version of Nakayama lemma applied to $\Omega_{B/A}$ in the almost category). Taking $A=K^{\circ a}$ or $A=K^{\flat\circ a}$ 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 $K$ and $K^{\circ a}$. In particular, the essential image of (1) consists of finite etale algebras $A/K$ such that $A^{\circ a}/K^{\circ a}$ is finite etale. To show the equivalence (1) and (4), it suffices to show that

Theorem 3 (Almost purity) Let $K$ be a perfectoid field. If $A/K$ is a finite etale algebra, then $A^{\circ a}/K^{\circ a}$ is finite etale.

Step 3 Show the almost purity for perfectoid fields $K$ of characteristic $p>0$ (hence (4) is an equivalence). This is not difficult by the existence of Frobenius. Here is the outline of the argument. Suppose $e\in A \otimes_K A$ is the idempotent from the finite etale algebra $A/K$, then there exists some integer $N$ such that $\varpi^N e\in A^{\circ}\otimes_{K^{\circ}}A^{\circ}$. Since the Frobenius is surjective, $\varpi^{N/p^n}e\in A^{\circ}\otimes_{K^{\circ}}A^{\circ}$ as well for any $n$. Since $n$ can be arbitrarily large, we obtain an almost idempotent for $A^{\circ}/K^{\circ}$ as desired.

Step 4 Show the almost purity for perfectoid fields $K$ 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.

References

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

[2]Gabber, O. and Ramero, L., Almost ring theory - sixth release, ArXiv Mathematics e-prints (2002).

[3]Fontaine, Jean-Marc and Wintenberger, Jean-Pierre, 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]Faltings, Gerd, Almost étale extensions, Astérisque (2002), no.279, 185--270.