Rigid Analytic Spaces

This blog post is inspired by and follows closely an amazing talk given by Ashwin Iyengar at the “What is a…seminar?” online seminar. I hope I can do the talk, and this wonderful topic, some justice in this blog post.

One of the most fascinating and powerful things about algebraic geometry is how closely tied it is to complex analysis, despite what the word “algebraic” might lead one to think. To state one of the more simple and common examples, we have that smooth projective curves over the complex numbers \mathbb{C} are the same thing as compact Riemann surfaces. In higher dimensions we also have Chow’s theorem, which tells us that an analytic subspace of complex projective space which is topologically closed is an algebraic subvariety.

This is all encapsulated in what is known as “GAGA“, named after the foundational work “Géometrie Algébrique et Géométrie Analytique” by Jean-Pierre Serre. We refer to the references at the end of this post for the more precise statement, but for now let us think of GAGA as giving us a fully faithful functor from proper algebraic varieties over \mathbb{C} to complex analytic spaces, which gives us an equivalence of categories between their coherent sheaves.

One may now ask if we can do something similar with the p-adic numbers \mathbb{Q}_{p} (or more generally an extension K of \mathbb{Q}_{p} that is complete with respect to a valuation that extends the one on \mathbb{Q}_{p}) instead of \mathbb{C}. This leads us to the theory of rigid analytic spaces, which was originally developed by John Tate to study a p-adic version of the idea (also called “uniformization”) that elliptic curves over \mathbb{C} can be described as lattices on \mathbb{C}.

Let us start defining these rigid analytic spaces. If we simply naively try to mimic the definition of complex analytic manifolds by having these rigid analytic spaces be locally isomorphic to \mathbb{Q}_{p}^{m}, with analytic transition maps, we will run into trouble because of the peculiar geometric properties of the p-adic numbers – in particular, as a topological space, the p-adic numbers are totally disconnected!

To fix this, we cannot just use the naive way because the notion of “local” would just be too “small”, in a way. We will take a cue from algebraic geometry so that we can use the notion of a Grothendieck topology to fix what would be issues if we were to just use the topology that comes from the p-adic numbers.

The Tate algebra \mathbb{Q}_{p}\langle T_{1},\ldots,T_{n}\rangle is the algebra formed by power series in n variables that converge on the n-dimensional unit polydisc D^{n}, which is the set of all n-tuples (c_{1},\ldots,c_{n}) of elements of \mathbb{Q}_{p} that have p-adic absolute value less than or equal to 1 for all i from 1 to n.

There is another way to define the Tate algebra, using the property of power series with coefficients in p-adic numbers that it converges on the unit polydisc D^{n} if and only if its coefficients go to zero (this is not true for real numbers!). More precisely if we have a power series

\displaystyle f(T_{1},\ldots,T_{n})=\sum_{a} c_{a}T_{1}^{a_{1}}\ldots T_{n}^{a_{n}}

where c_{a}\in \mathbb{Q}_{p} and a=a_{1}+\ldots+a_{n} runs over all n-tuples of natural numbers, then f converges on the unit polydisc D^{n} if and only if \lim_{a\to 0}c_{a}=0.

The Tate algebra has many important properties, for example it is a Banach space (see also Metric, Norm, and Inner Product) with the norm of an element given by taking the biggest p-adic absolute value among its coefficients. Another property that will be very important in this post is that it satisfies a Nullstellensatz – orbits of the absolute Galois group \mathrm{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p}) in D^{n} correspond to maximal ideals of the Tate algebra. Explicitly, this correspondence is given by the evaluation map x\in D^{n}\mapsto \lbrace f\vert f(x)=0\rbrace.

A quotient of the Tate algebra by some ideal is referred to as an affinoid algebra. The maximal ideals of the underlying set of an affinoid algebra A will be denoted \mathrm{Max}(A), and this should be reminiscent of how we obtain the closed points of a scheme.

Once we have the underlying set \mathrm{Max}(A) , we need two other things to be able to define “affinoid” rigid analytic spaces, which we shall later “glue” to form more general rigid analytic spaces – a topology, and a structure sheaf (again this should be reminiscent of how schemes are defined).

Again taking a cue from how schemes are defined, we define “rational domains”, which are analogous to the distinguished Zariski open sets of (the underlying topological space of) a scheme. Given elements f_{1},\ldots,f_{r},g of the affinoid algebra A, the rational domain \displaystyle A\left(\frac{f}{g}\right) is the set of all x\in\mathrm{Max}(A) such that f_{i}(x)\leq g(x) for all 1\leq i\leq r.

The rational domains generate a topology, however this is still just the p-adic topology, and so it still does not solve the problems that we originally ran into when we tried to define rigid analytic spaces by mimicking the definition of a complex analytic manifold. The trick will be to make use of something that is more general than just a topology – a Grothendieck topology (see also More Category Theory: The Grothendieck Topos).

Let us now define the particular Grothendieck topology that we will use. Unlike other examples of a Grothendieck toplogy, the covers will involve only subsets of the space being covered (it is also referred to as a mild Grothendieck topology). Let X=\mathrm{Max}(A). A subset U of X is called an admissible open if it can be covered by rational domains \lbrace U_{i}\rbrace_{i\in I} such that for any map Y\to X where Y=\mathrm{Max}(B) for some affinoid algebra B, the covering of Y given by the inverse images of the U_{i}‘s admit a finite subcover.

If U is an admissible open covered by admissible opens \lbrace U_{i}\rbrace_{i\in I}, then this covering is called admissible if for any map Y\to X whose image is contained in U, the covering of Y given by the inverse images of the U_{i}‘s admit a finite subcover. These admissible coverings satisfy the axioms for a Grothendieck topology, which we denote G_{X}.

If A is an affinoid algebra, and f_{1},\ldots,f_{k},g are functions, we let \displaystyle A\left\langle \frac{f}{g}\right\rangle denote the ring A\langle T_{1},\ldots T_{k}\rangle/(gT_{i}-f_{i}). By associating to a rational domain \displaystyle A\left(\frac{f}{g}\right) this ring \displaystyle A\left\langle\frac{f}{g}\right\rangle, we can define a structure sheaf \cal{O}_{X} on this Grothendieck topology.

The data consisting of the set X=\mathrm{Max}(A), the Grothendieck topology G_{X}, and the structure sheaf \mathcal{O}_{X} is what makes up an affinoid rigid analytic space. Finally, just as with schemes, we define a more general rigid analytic space as the data consisting of some set X, a Grothendieck topology G_{X} and a sheaf \mathcal{O}_{X} such that locally, with respect to the Grothendieck topology G_{X}, it is isomorphic to an affinoid rigid analytic space.

Under this definition, we have in fact a version of GAGA that holds for rigid analytic spaces – a fully faithful functor from proper schemes of finite type over \mathbb{Q}_{p} to rigid analytic spaces over \mathbb{Q}_{p} that gives an equivalence of categories between their coherent sheaves.

Finally let us now look at an example. Consider the affinoid rigid analytic space obtained from the affinoid algebra \mathbb{Q}_{p}\langle T\rangle. By the Nullstellensatz the underlying set is the unit disc D. The “boundary” of this is the rational subdomain (and therefore an admissible open) \displaystyle D\left(\frac{1}{T}\right), and its complement, the “interior” is covered by rational subdomains \displaystyle D\left(\frac{T^{n}}{p}\right). With this covering the interior may also be shown to be an admissible open.

While it appears that, since we found two complementary admissible opens, we can disconnect the unit disc, we cannot actually do this in the Grothendieck topology, because the set consisting of the boundary and the interior is not an admissible open! And so in this way we see that the Grothendieck topology is the difference maker that allows us to overcome the obstacles posed by the peculiar geometry of the p-adic numbers.

Since Tate’s innovation, the idea of a p-adic or nonarchimedean geometry has blossomed with many kinds of “spaces” other than the rigid analytic spaces of Tate, for example adic spaces, a special class of which generalize perfectoid fields (see also Perfectoid Fields) to spaces, or Berkovich spaces, which are honest to goodness topological spaces instead of relying on a Grothendieck topology like rigid analytic spaces do. Such spaces will be discussed on this blog in the future.


Rigid analytic space on Wikipedia

Algebraic geometry and analytic geometry on Wikipedia

Several approaches to non-archimedean geometry by Brian Conrad

Reciprocity laws and Galois representations: recent breakthroughs by Jared Weinstein

p-adic families of modular forms [after Coleman, Hida, and Mazur] by Matthew Emerton