Perfectoid uniformization, part one (ed.: and part two)

Let C be a complete algebraically closed extension of \mathbf{Q}_p, and let X be a nice rigid analytic space/locally Noetherian adic space over C.  What should it mean for X to be covered by a perfectoid space?  Here’s one possible definition:

Definition. perfectoid covering space Y of X is a perfectoid space Y \in X_{\mathrm{proet}} with the following property:

  • Locally on some covering X = \cup U_i of X by connected affinoids U_i, the map f: Y \to X can be factored as Y_i := f^{-1}(U_i) \to Z_i \to U_i, where Z_i \to U_i is an etale covering map in de Jong’s sense and Y_i \to Z_i realizes Y_i as a perfectoid object in (Z_i)_{\mathrm{profet}}.

Note that Y_i has an induced structure as a perfectoid space, so the final condition makes sense. Note also that each Z_i is still rigid analytic/locally Noetherian/whatever, and that each connected component of Z_i surjects onto U_i (this is slightly nontrivial if X isn’t normal). If you don’t know about etale covering maps, you should go read the paper where de Jong defines them, “Etale fundamental groups of nonarchimedean analytic spaces” – you won’t regret it.  I should also note that the connected components of Z_i need not be affinoids!  This is one of those funky twists in the nonarchimedean universe: one might think that “connected and locally finite-etale-surjective with connected affinoid target” would imply affinoid, and yet it doesn’t.

OK, what should it mean for such a covering to be Galois?

Definition. A perfectoid covering space Y/X is Galois with Galois group G:= \mathrm{Aut}(Y/X) if the following two conditions are true:

  • The fiber of the map Y \to X over any geometric point of X is a G-torsor.
  • Make G into a topological group by declaring that stabilizers of open affinoid perfectoid subsets of Y give a basis of open subgroups. Then Y \times_{X} Y is homeomorphic to G \times Y.

If X is qcqs then I’d guess that G is always locally profinite, but I haven’t tried to check this. In any case, in the examples below G will always be locally profinite and it’ll be clear what to do. For a more careful treatment of some of these things, take a look at Peter’s recent paper on the Lubin-Tate tower.

Basic example. Take X= \mathbf{A}^1, the rigid analytic affine line over C, and take Z = (\mathrm{Spf}(\mathcal{O}_C [[T]]))_{\eta} to be the open unit disk over C. The map Z \to X sending x to \log (1+x) is the primordial example of an etale covering map: it is of infinite degree, with connected source and target, and Galois with Galois group \mu_{p^{\infty}} (which acts by sending T to \zeta (1+T)-1).
Now, living over Z we have the perfectoid covering Y given by adjoining all p-power roots of 1+T, i.e. Y \sim \lim_{\leftarrow \varphi} Z where \varphi acts via T \mapsto (1+T)^p -1. (Exercise: Write Y as the generic fiber of a suitable \mathrm{Spf}.) This is Galois over Y with Galois group \mathbf{Z}_p (1). Putting these maps together, we get:
The map Y \to \mathbf{A}^1 realizes Y as a perfectoid covering space of \mathbf{A}^1, and this covering is connected and Galois with Galois group \mathbf{Q}_p(1) \simeq \mathbf{Q}_p.

Next time, we’re going to prove the following theorem, which (to me at least) is very surprising, and even somewhat disturbing:

Theorem. Let X be the analytic space associated with a normal projective variety over C. Then one can find a perfectoid covering space \tilde{X}/X with the following properties:

  • The covering \tilde{X}/X is connected and Galois with locally profinite Galois group G, and X^{\lozenge} \cong \tilde{X}^{\lozenge} / G as diamonds over \mathrm{Spd}\,C. In particular, X can be recovered from the data of \tilde{X} with its action of G.
  • The space \tilde{X} is quasi-Stein in the sense that it has a covering by a rising union V_1 \subset V_2 \subset \cdots \subset V_i \subset \cdots of affinoid perfectoid spaces.

So every normal projective variety can be uniformized by a “contractible” perfectoid space! This is wildly divergent from the situation over \mathbf{C}.

Edit (8/9/2015): It turns out I’m too lazy to carefully write up a proof of this theorem right now, so here’s a quick sketch of the idea.  One treats the case of X= \mathbf{P}^n directly, taking \tilde{X} to be a piece of the Lubin-Tate tower at infinite level, and the claimed properties of \tilde{X} follow from Jared and Peter’s explicit description of the Lubin-Tate space at infinite level; in this case, one ends up with G= \mathrm{SL}_{n+1}(\mathbf{Q}_p).  The general case follows from taking a projective embedding, and using the fact that closed immersions into perfectoid spaces are perfectoid.

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