Let be a complete algebraically closed extension of , and let be a nice rigid analytic space/locally Noetherian adic space over . What should it mean for to be covered by a perfectoid space? Here’s one possible definition:
Definition. A perfectoid covering space of is a perfectoid space with the following property:
- Locally on some covering of by connected affinoids , the map can be factored as , where is an etale covering map in de Jong’s sense and realizes as a perfectoid object in .
Note that has an induced structure as a perfectoid space, so the final condition makes sense. Note also that each is still rigid analytic/locally Noetherian/whatever, and that each connected component of surjects onto (this is slightly nontrivial if 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 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 is Galois with Galois group if the following two conditions are true:
- The fiber of the map over any geometric point of is a -torsor.
- Make into a topological group by declaring that stabilizers of open affinoid perfectoid subsets of give a basis of open subgroups. Then is homeomorphic to .
If is qcqs then I’d guess that is always locally profinite, but I haven’t tried to check this. In any case, in the examples below 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 , the rigid analytic affine line over , and take to be the open unit disk over . The map sending to is the primordial example of an etale covering map: it is of infinite degree, with connected source and target, and Galois with Galois group (which acts by sending to ).
Now, living over we have the perfectoid covering given by adjoining all -power roots of , i.e. where acts via . (Exercise: Write as the generic fiber of a suitable .) This is Galois over with Galois group . Putting these maps together, we get:
The map realizes as a perfectoid covering space of , and this covering is connected and Galois with Galois group .
Next time, we’re going to prove the following theorem, which (to me at least) is very surprising, and even somewhat disturbing:
Theorem. Let be the analytic space associated with a normal projective variety over . Then one can find a perfectoid covering space with the following properties:
- The covering is connected and Galois with locally profinite Galois group , and as diamonds over . In particular, can be recovered from the data of with its action of .
- The space is quasi-Stein in the sense that it has a covering by a rising union 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 .
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 directly, taking to be a piece of the Lubin-Tate tower at infinite level, and the claimed properties of follow from Jared and Peter’s explicit description of the Lubin-Tate space at infinite level; in this case, one ends up with . The general case follows from taking a projective embedding, and using the fact that closed immersions into perfectoid spaces are perfectoid.