A Motivation

\(\newcommand{\a}{\alpha} \newcommand{\C}{\mathbb{C}} \newcommand{\P}{\mathbb{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\X}{\mathfrak{X}} \newcommand{\Ox}{\mathcal{O}} \newcommand{\R}{\mathbb{R}}\) To see what formal geometry attempts to do, we might first want to understand where “classical” algebraic geometry has its shortcomings. So while the world of schemes is very useful, because of its great generality, there is a toll to be paid when we want to focus on more special kinds of schemes. For example, complex projective varieties, so e.g. \(\mathbb{P}^n_{\mathbb{C}}\) as the complex manifold.

By Serre’s GAGA, every complex projective variety arises from a projective scheme \(X/\mathbb{C}\), in the sense that it will be isomorphic to the \(\mathbb{C}\)-points \(X(\mathbb{C})\). Similar statements also exist for coherent sheaves. So complex geometry is not too far away from algebraic geometry… But wait! From the algebraic geometry point of view \(X(\mathbb{C})\) only sees the Zariski topology, but of course \(X(\mathbb{C})\) also naturally has the complex topology, which is finer. Heuristically, we can hope that this will allow us to make stronger statements, using exactly this additional structure. Here are two instances, where working over \(\mathbb{C}\) is beneficial.

Example 1

Hodge theory! Let \(H^n(X(\mathbb{C}),\mathbb{C})\) be either the singular cohomology, or the de Rham cohomology, the former crucially using the complex topology, while the latter has an algebraic analogue, the algebraic de Rham cohomology. In any case, the concise way to state the Hodge theorem is as follows

Theorem Let \(X\) be as above and smooth. Then we have a natural decomposition (as \(\mathbb{C}\) vector spaces)

\[H^n(X(\mathbb{C}),\mathbb{C})\cong\bigoplus_{p+q=n} H^{p,q}(X(\mathbb{C}))\]

and furthermore

\[H^{p,q}(X(\mathbb{C}))\cong \overline{ H^{q,p}(X(\mathbb{C}))}.\]

The groups \(H^{p,q}(X(\mathbb{C}))\) can be defined as Dolbeaut cohomology groups, but one can also think of them as \(H^{p,q}(X(\mathbb{C}))\cong H^p(X,\Omega^q)\). Now to be fair the first part of this theorem can be proved also in a more general set up, by showing that some spectral sequence degenerates. But then the decomposition will be not canonical and more importantly the second part of theorem, the Hodge symmetry, is unique to the \(\mathbb{C}\) case. One way to prove the above theorem that I am aware of is presented in the book by Voision [ref] and uses harmonic analysis.

Example 2

The second example uses the exponential function. To be more precise we have the exponential short exact sequence

\[0\to \underline{\mathbb{Z}}\xrightarrow{\cdot 2\pi i} \mathcal{O}\xrightarrow{exp( \_ )} \mathcal{O}^*\to 0.\]

This induces a long exact sequence, which should be familiar with those who studied some complex geometry/curves

\[...H^1(X(\C),\Ox)\to H^1(X(\C),\Ox^*)\to H^2(X(\C),\underline{\Z})\to...\]

The above examples can be further elaborated, and there are more examples of course. But let us now proceed to our actual subject of interest.

What happens, if we want to work over other fields than just \(\C\) or \(\R\). Just as \(\R\) is nothing but the completion of \(\Q\) for the euclidean topology, we might wonder what happens if we consider varieties over \(\Q_p\), or its algebraic closure, or its completion. Again since these fields have more structure, a non-trivial topology namely, we can hope for a theory of something like \(p\)-adic geometry (in constrast to complex geometry). Put in simple terms this is one thing formal geometry attempts to do (or maybe considering more general kind of topological ring). Unfortunately things turn out to be not as easy as in the complex case. So far there is no “one construction” that is fit best for all purposes. Each approach, that will be briefly introduced in the next section, resolves certain issues, but in return imposes restrictions in a different way.

One of the first things that clearly distinguishes the \(p\)-adic topology form the euclidean topology are the closed and open unit disk

\[\Ox_K=\{x\in K \mid \lVert x \rVert \leq 1 \} \ , \ m_K=\{x\in K\mid \lVert x \rVert < 1 \},\]

this is now a local ring! So to liberate us from the real world, we have to come up with new techniques and rethink the notions we are familiar with. Throughout, the unit disk will follow us as an essential example, as a toy case, to which we can apply to our new constructions.

A map of formal geometry

Since a map is always only an approximation of the territory that it encaptures, what I will write is only a sketch of the whole area and as such to be read with an appropriate amount of caution.

Before introducing the different approaches, what are reasonable things to expect from them?

Ideally we end up with a fully faithful functor:
\[S:\{\textrm{rings with a topology+...}\}^{op}\to \{\textrm{???-spaces}\}\]
as an analogue of
\[Spec:\{\textrm{rings}\}^{op}\to \{\textrm{Locally ringed spaces}\}.\]
So the objects we construct should live in a reasonable ambient category, e.g. it should be a topological space with a sheaf. I am intentionally vague with the domain category of the functor, as it will vary from approach to approach.
Now let’s say we have \(S(A)\), where \(A\) is an object in our domain category. \(S(A)\) will have a topology, sometimes also natural Grothendieck topologies. Let \(f,g\in A\) be some elements. The following sets should be open:
\[\{x\in S(A)\mid f(x)\neq 0 \}\]
analogous to the scheme case. It again remains vague what I mean here e.g. by \(f(x)\), but the usual case of affine schemes should provide some intuition. Now furthermore we would of course like to include more open sets! The straightforward approach would be something like
\[\{x\in S(A)\mid |f(x)| \leq 1 \}\]
or if we are feeling very bold, maybe more generally
\[\{x\in S(A)\mid |f(x)| \leq |g(x)| \}.\]
On the other hand if \(I\subset A\) is a “nice” ideal, then something like
\[V(I)=\{x\in S(A)\mid f(x)=0 \forall f\in I\}\]
should be closed in \(S(A)\), maybe something like \(S(A/I)\hookrightarrow S(A)\).
Having talked about topology, it seems reasonable to bring up the subject of gluing. Just like with schemes, in our category of \(\textrm{???-spaces}\) we should be able to take some kind of “gluing” datum and get a new ???-space, which of course locally looks like \(S(A)\).
It would be nice to have things like fibre products and other kinds of (co)limits. Especially, if our domain category has some kind of (co)limit, the functor \(S\) should preserve it.

Approaches

But as mentioned before, we always will have to make concessions. With this in mind and without further ado, let us briefly talk about the approaches:

Formal schemes
Rigid analytic spaces
Berkovich (k-analytic) spaces
Adic spaces

And all of them are closely interwined, as we will see in the next posts. The following will be a short informal overview

1. Formal schemes

The approach that formal schemes take is, as the name indicates, very close to the theory of schemes. It is thus very general and one can define it basically for any kind of topological ring. Furthermore, formal schemes contain the category of schemes (i.e. there is a fully faithful functor from Schemes to Formal schemes). This becomes more clear, once the essential idea of formal schemes is introduced: Instead of looking at the spectrum \(Spec(A)\), the set of prime ideals, we will consider the formal spectrum \(Spf(A)\), the set of open prime ideals, for a topological ring \(A\) (these two are the same, if \(A\) carries the discrete topology and this is how you can consider each scheme as formal scheme). Now while formal schemes are somewhat tame, as they are close to schemes, they are maybe a bit too close to schemes. In particular it is too coarse to capture the finer subtleties of \(p\)-adic geometry. This leads us to…

2. Rigid analytic spaces

In some sense this is the opposite approach. Instead of covering a broad class of objects, here we focus on the concept of “\(p\)-adic variety”. What is a (classical) affine variety, but a \(k\)-algebra of finite type, which is of them form \(k[x_1,...,x_n]/I\), for some ideal \(I\). In this spirit, affine rigid analytic spaces over a non-archimedean field \(K\), correspond to \(K\)-algebras topologically of finite type, which are of the form \(A=K<x_1,...,x_n>/I\), for some ideal \(I\) (which is automatically closed). Here \(K<x_1,...,x_n>\) is the Tate algebra over \(K\). It serves as analogue of the polynomial ring and is supposed to represent the algebra of convergent power series on the unit disk, but more on that later. Now consider \(Sp(A)\), which is the set maximal ideals. Great, this is very intuitive and close to the complex case at first sight. But we get into some trouble, when we want to endow \(Sp(A)\) with a well-behaved topology (as mentioned above), let alone a structure sheaf. Sadly, the Zariski topology won’t do the job. This is why in the end one has to define a \(G\)-topology (which is short for Grothendieck topology) and talk about admissible covers. To fix this, one can try to add more points to \(Sp(A)\) in reasonable way, so we have to consider…

3. Berkovich spaces

Motivated by rigid analytic varieties, we want to fix the issue just mentioned above. Let us again take such an \(A\) as above. One way to add points would be of course to consider \(Spec(A)\), but this will make everything too algebraic again. Instead one considers another kind of spectrum, \(M(A)\), which is the set of bounded multiplicative seminnorms. One can show that each maximal ideal uniquely defines such a seminorm, i.e. we can embedd \(Sp(A)\) into \(M(A)\). But now we have new kinds of “generic” points, most notably in the case of the unit disk, we get the Gauss point. This will already resolve a few issues with the topology, it will be in fact better behaved. But, for reasons shown later, actually one can add even more points in a natural way, so we end up with…

4. Adic spaces

Adic spaces, and this can already be seen as a first kind of complication, are not defined in terms of a ring \(A\) with a topology, but in terms of pairs of rings \((A,A^+)\), where we can think of \(A^+\subset A\), in some instances at least, as subring of bounded elements in \(A\). The most canonical example might be \((K,\Ox_K)\). The construction of the adic space \(Spa(A,A^+)\) is similar to the one above and is in terms of valuations (or rather equivalence classes of such). But allowing more general valuations (to be specified later), one gets even more points. While this approach has been used heavily recently, it has some rather horrid behaviour, e.g. when it comes to fibre products, as they might not even exist! And here we might end up in a circular loop, as formal schemes behave much better in terms of (co)limits than adic spaces.