# Perfectoid Fields

Consider the field of $p$-adic numbers $\mathbb{Q}_{p}$. An element of $\mathbb{Q}_{p}$ may be written in the form

$\displaystyle \sum_{n=k}^{\infty}a_{n}p^{n}$

with each $a_{n}$ being an element of the finite field $\mathbb{F}_{p}$. Let us compare this with the field of Laurent series $\mathbb{F}_{p}((t))$ in one variable $t$ over $\mathbb{F}_{p}$. An element of $\mathbb{F}_{p}((t))$ may be written in the form

$\displaystyle \sum_{m=l}^{\infty}a_{m}t^{m}$

We see that they look very similar, even though $\mathbb{Q}_{p}$ is characteristic $0$, and $\mathbb{F}_{p}((t))$ is characteristic $p$.

How far can we push this analogy? The fact that one is in characteristic $0$, and the other is characteristic $p$ means we cannot ask for an isomorphism of fields. However, the Fontaine-Wintenberger theorem gives us another connection between $\mathbb{Q}_{p}$ and $\mathbb{F}_{p}((t))$ – if we modify them by adjoining $p$-power roots of $p$ and $t$ respectively. This theorem states that the fields $\cup_{n}\mathbb{Q}_{p}(p^{1/p^{n}})$ and $\cup_{n}\mathbb{F}_{p}((t^{1/p^{n}}))$ have the same absolute Galois group! By the fundamental theorem of Galois theory, this means the category formed by their extensions will be equivalent as well.

We now let $F$ denote the completion of $\cup_{n}\mathbb{Q}_{p}(p^{1/p^{n}})$, and we let $F^{\flat}$ suggestively denote the completion of $\cup_{n}\mathbb{F}_{p}((t^{1/p^{n}}))$. Completing these fields does not change their absolute Galois groups, so the absolute Galois groups of $F$ and $F^{\flat}$ remain isomorphic. We say that the characteristic $p$ field $F^{\flat}$ is the tilt of the characteristic $0$ field $F$, and that $F^{\flat}$ is an untilt of $F$ (note the subtle change in our choice of article – untilts are not unique).

In this post, we will explore these kinds of fields – which are called perfectoid fields – and the process of tilting and untilting that bridges the world of characteristic $0$ and characteristic $p$. After Fontaine and Wintenberger came up with their famous theorem their ideas have since been developed into even more general and even more powerful theories of perfectoid rings and perfectoid spaces – but we will leave these to future posts. For now we concentrate on the case of fields.

First let us look at a much more primitive example of bridging the world of characteristic $0$ and characteristic $p$. Consider $\mathbb{Q}_{p}$ (characteristic $0$). It has a ring of integers $\mathbb{Z}_{p}$, whose residue field is $\mathbb{F}_{p}$ (characteristic $p$). To got the other way, starting from $\mathbb{F}_{p}$ we can take its ring of Witt vectors, which is $\mathbb{Z}_{p}$. Then we take its field of fractions which is $\mathbb{Q}_{p}$.

More generally, there is a correspondence between characteristic $0$ discretely valued complete fields whose uniformizer is $p$ and characteristic $p$ fields which are perfect, i.e. for which the Frobenius morphism is bijective, and the way to go from one category to the other is as in the previous paragraph.

This is a template for “bridging the world of characteristic $0$ and characteristic $p$“. However, we may want more, something like the Fontaine-Wintenberger theorem where the characteristic $0$ object and the characteristic $p$ object have isomorphic absolute Galois groups. We will be tweaking this basic bridge in order to create something like Fontaine-Winterger theorem, and these tweaks will lead us to the notion of a perfectoid field. However, we already have isolated one property that we want from such a “perfectoid” field:

The first property that we want from a perfectoid field is that it has to be nonarchimedean. This allows us to have a “ring of integers” that serves as an intermediary object between the two worlds, as we have seen above.

Now let us concentrate on the Fontaine-Wintenberger theorem. To understand this phenomenon better, we need to make use of a version of the fundamental theorem of Galois theory, which allows us to think in terms of extensions of fields instead of their Galois groups. More properly, we want an equivalence of categories between the “Galois categories” of certain extensions of these “base” fields and this will be the property of these base fields being perfectoid. Now the problem is that the extensions that we are considering may not fit into the primitive correspondence we stated above – for example the corresponding characteristic $p$ object may not be perfect, i.e. the Frobenius morphism may not be surjective.

The fix to this is a kind of “perfection”, which is the tilting functor we mentioned earlier. Let $R$ be a ring. The tilt of $R$, denoted $R^{\flat}$ is defined to be the inverse limit

$\displaystyle R^{\flat}=\varprojlim_{x\mapsto x^{p}}R/pR$

In other words, an element $x$ of $R^{\flat}$ is an infinite sequence of elements $(x_{0},x_{1},x_{2},\ldots)$ of the quotient $R/pR$ such that $x_{1}\cong x_{0}^{p}\mod p$, $x_{2}\cong x_{1}^{p}\mod p$, and so on. We want $R^{\flat}$ to be a ring, so we define it to have componentwise multiplication, i.e.

$\displaystyle (xy)_{i}=x_{i}y_{i}$

However the addition is going to be more complicated. We define it, for each component, as follows:

$\displaystyle (x+y)_{i}=\lim_{n\to\infty}(x_{i+n}+y_{i+n})^{p^n}$

At this point we take the opportunity to define another important concept in the theory of perfectoid fields (and rings). Let $W$ be the Witt vector functor (see also The Field with One Element). Then we give the Witt vectors of the tilt of $R$, $W(R^{\flat})$, a special name. We will refer to this ring as $A_{\mathrm{inf}}(R)$. It will make an appearance again later. For now we note that there is going to be a canonical map $\theta: A_{\mathrm{inf}}(R)\to R$.

As we can see, we have defined the tilt of an arbitrary ring. This is not exclusive to the ones which are “perfectoid” whatever the definition of “perfectoid” may be (we will come to this later of course). Again what makes perfectoid fields (such as our earlier examples) special though, is that if $F$ is a perfectoid field of characteristic $0$, then $F$ and its tilt $F^{\flat}$ will have isomorphic absolute Galois groups. This will actually follow from the following statement (together with some technicalities involving fiber functors and so on):

There is an equivalence of categories between the category of finite etale algebras over a perfectoid field $F$ and the category of finite etale algebras over its tilt $F^{\flat}$.

This in turn will follow from the following two statements:

1. Finite extensions of perfectoid fields are perfectoid.
2. There is an equivalence of categories between the category of perfectoid extensions of a perfectoid field $F$ and the category of perfectoid extensions over its tilt $F^{\flat}$.

This equivalence of categories is given by tilting a perfectoid extension over $F$. This will actually give us a perfectoid extension over $F^{\flat}$. However, we need a functor that goes in the other direction, a “quasi-inverse” that when composed with tilting gives us back our original perfectoid extension over $F$ (or at least something isomorphic to it, this is what the “quasi-” part means). However, we also said in an earlier paragraph that the “untilt” of a characteristic $p$ field may not be unique (two different untilts may also not be isomorphic). How do we approach this problem?

We recall again the ring $A_{\mathrm{inf}}(R)$ defined earlier as the ring of Witt vectors of the tilt of $R$, and we recall that it has a canonical map $\theta:A_{\mathrm{inf}}(R)\to R$. If we know this map, and if we know that it is surjective, then we can recover $R$ simply by quotienting out by the kernel of the map $\theta$!

The problem is that (aside from not knowing whether it is in fact surjective or not) is that we only know this map if we know that $R^{\flat}$ was obtained as the tilt of $R$. If we were simply handed some characteristic $p$ field for instance we would not be able to know this map.

However, note that we are interested in an equivalence of categories between the category of perfectoid extensions over the field $F$ and the corresponding category over its tilt $F^{\flat}$. By specifying these “bases” $F$ and $F^{\flat}$, it is in fact enough to specify unique untilts! In other words, if we have say just some perfectoid field $A$, we cannot determine a unique untilt for it, but if we say in addition that it is a perfectoid extension over $F$, and we are looking for the unique untilt of it over $F^{\flat}$, we can in fact find it, as long as the map $\theta$ is surjective.

So now how do we guarantee that $\theta$ is surjective? This brings us to our second property, which is that the Frobenius morphism from $\overline{R}$ to itself must be surjective. This is actually the origin of the word “perfectoid”; since as above a field for which the Frobenius morphism is bijective is called perfect; hence, requiring it to be surjective is a relaxation of this condition. This condition guarantees that the map $\phi:A_{\mathrm{inf}}(R)\to R$ is going to be surjective.

The final property that we want from a perfectoid field is that its valuation must be non-discretely valued. The reason for this is that we want to consider infinitely ramified extensions of $\mathbb{Q}_{p}$. The two previous conditions that we want can only be found in unramified (discretely valued) or infinitely ramified (non-discretely valued) of $\mathbb{Q}_{p}$. We have already seen above that if we only look at the ones which are unramified then our corresponding characteristic $p$ objects will be limited to perfect $\mathbb{F}_{p}$-algebras, and this is not enough to give us the Fontaine-Wintenberger theorem. Therefore we will want infinitely ramified extension of $\mathbb{Q}_{p}$, and these are non-discretely valued.

These three properties are enough to give us the Fontaine-Wintenberger theorem. To summarize – a perfectoid field is a complete, nonarchimedean field $F$ such that the Frobenius morphism from $\mathcal{O}_{F}/\mathfrak{p}$ to itself is surjective and such that its valuation is non-discretely valued.

We have only attempted to motivate the definition of a perfectoid field in this post, and barely gone into any sort of detail. For that one can only recommend the excellent post by Alex Youcis on his blog The Fontaine-Wintenberger Theorem: Going Full Tilt, which inspired this post, but barely does it any justice.

Aside from the Fontaine-Wintenberger theorem, the concepts we have described here – the idea behind “perfectoid”, the equivalence of categories of perfectoid extensions that gives rise to the Fontaine-Wintenberger theorem, the idea of tilting and untilting which bridges the worlds of characteristic $0$ and characteristic $p$, the ring $A_{\mathrm{inf}}(R)$, and so on, have found much application in many areas of math, from the aforementioned perfectoid rings and perfectoid spaces, to p-adic Hodge theory, and to many others.

References:

Perfectoid Space on Wikipedia

What is…a Perfectoid Space? by Bhargav Bhatt

Berkeley Lectures on p-adic Geometry by Peter Scholze and Jared Weinstein

Reciprocity Laws and Galois Representations: Recent Breakthroughs by Jared Weinstein

The Fontaine-Wintenberger Theorem: Going Full Tilt by Alex Youcis