Report from Tucson

Just back from the 2017 Arizona Winter School on perfectoid spaces.  First of all, I should say that everything was impressively well-organized, and that the lecturers did a fantastic job, especially considering the technical weight of this material. (Watch the videos if you don’t believe me.)  Jared Weinstein, in particular, has an almost supernatural ability to make a lecture on some technical thing feel comforting.

Now to the jokes.

  • In his opening lecture, Scholze called perfectoid spaces a “failed theory”, on account of his inability to completely settle weight-monodromy. “You see, I’m Prussian, and when a Prussian says he wants to do something, he really feels responsible for doing it.”
  • Audience member: “Why are they called diamonds?”
    Scholze: “[oral explanation of the picture on p. 63 of the Berkeley notes]”
    Weinstein: “Also, diamonds are hard.”
  • Anon.: “When you’re organizing a conference, the important thing is not to give in and be the first one who actually does stuff.  Because then you’ll end up doing everything!  Don’t do that!  Don’t be the dumb one!”
    Me: “Didn’t you organize [redacted] a couple of years ago?”
    Anon.: “Yeah… It turned out that Guido Kings was the dumb one.”
  • Mazur: “It just feels like the foundations of this area aren’t yet… hmm…”
    Me: “Definitive?”
    Mazur: “Yes, exactly.  I mean, if Grothendieck were here, he would be screaming.”
  • “Do you ever need more than two legs?”
  • During the hike, someone sat on a cactus.
  • Finally, here is a late night cartoon of what a universal cohomology theory over \mathbb{Z} might look like (no prizes for guessing who drew this):

What does an inadmissible locus look like?

Let H/ \overline{\mathbf{F}_p} be some p-divisible group of dimension d and height h, and let \mathcal{M} be the rigid generic fiber (over \mathrm{Spa}\,\breve{\mathbf{Q}}_p) of the associated Rapoport-Zink space. This comes with its Grothendieck-Messing period map \pi: \mathcal{M} \to \mathrm{Gr}(d,h), where \mathrm{Gr}(d,h) is the rigid analytic Grassmannian paramatrizing rank d quotients of the (covariant) rational Dieudonne module M(H) /\breve{\mathbf{Q}}_p. Note that \mathrm{Gr}(d,h) is a very nice space: it’s a smooth connected homogeneous rigid analytic variety, of dimension d(h-d).

The morphism \pi is etale and partially proper (i.e. without boundary in Berkovich’s sense), and so the image of \pi is an open and partially proper subspace* of the Grassmannian, which is usually known as the admissible locus. Let’s denote this locus by \mathrm{Gr}(d,h)^a. The structure of the admissible locus is understood in very few cases, and getting a handle on it more generally is a famous and difficult problem first raised by Grothendieck (cf. the Remarques on p. 435 of his 1970 ICM article). About all we know so far is the following:

  • When d=1 (so \mathrm{Gr}(d,h) = \mathbf{P}^{h-1}) and H is connected, we’re in the much-studied Lubin-Tate situation. Here, Gross and Hopkins famously proved that \pi is surjective, not just on classical rigid points but on all adic points, so \mathrm{Gr}(d,h)^a = \mathrm{Gr}(d,h) is the whole space. This case (along with the “dual” case where h>2,d=h-1) turns out to be the only case where \mathrm{Gr}(d,h)^a = \mathrm{Gr}(d,h), cf. Rapoport’s appendix to Scholze’s paper on the Lubin-Tate tower.
  • When H \simeq \mathbf{G}_m^{d} \oplus (\mathbf{Q}_p/\mathbf{Z}_p)^{h-d}, i.e. when H has no bi-infinitesimal component, it turns out that \mathrm{Gr}(d,h)^a = \mathbf{A}^{d(h-d)} is isomorphic to rigid analytic affine space of the appropriate dimension, and can be identified with the open Bruhat cell inside \mathrm{Gr}(d,h). This goes back to Dwork, who proved it when d=1,h=2. (I don’t know a citation for the general result, but presumably for arbitrary d,h this is morally due to Serre-Tate/Katz?)
  • In general there’s also the so-called weakly admissible locus \mathrm{Gr}(d,h)^{wa} \subset \mathrm{Gr}(d,h), which contains the admissible locus and is defined in some fairly explicit way. It’s also characterized as the maximal admissible open subset of the Grassmannian with the same classical points as the admissible locus. In the classical rigid language, the map \mathrm{Gr}(d,h)^a \to \mathrm{Gr}(d,h)^{wa} is etale and bijective; this is the terminology used e.g. in Rapoport-Zink’s book.
  • In general, the admissible and weakly admissible loci are very different.  For example, when H is isoclinic and (d,h)=1 (i.e. when M(H) is irreducible as a \varphi-module), \mathrm{Gr}(d,h)^a contains every classical point, and \mathrm{Gr}(d,h)^{wa} = \mathrm{Gr}(d,h), so the weakly admissible locus tells you zilch about the admissible locus in this situation (and they really are different for any 1 < d < h-1).

That’s about it for general results.

To go further, let’s switch our perspective a little. Since \mathrm{Gr}(d,h)^a is an open and partially proper subspace of \mathrm{Gr}(d,h), the subset |\mathrm{Gr}(d,h)^a| \subseteq |\mathrm{Gr}(d,h)| is open and specializing, so its complement is closed and generalizing.  Now, according to a very general theorem of Scholze, namely Theorem 2.42 here (for future readers, in case the numbering there changes: it’s the main theorem in the section entitled “The miracle theorems”), if \mathcal{D} is any diamond and E \subset |\mathcal{D}| is any locally closed generalizing subset, there is a functorially associated subdiamond \mathcal{E} \subset \mathcal{D} with |\mathcal{E}| = E inside |\mathcal{D}|. More colloquially, one can “diamondize” any locally closed generalizing subset of |\mathcal{D}|, just as any locally closed subspace of |X| for a scheme X comes from a unique (reduced) subscheme of X.

Definition. The inadmissible/nonadmissible locus \mathrm{Gr}(d,h)^{na} is the subdiamond of \mathrm{Gr}(d,h)^{\lozenge} obtained by diamondizing the topological complement of the admissible locus, i.e. by diamondizing the closed generalizing subset |\mathrm{Gr}(d,h)^a|^c \subset |\mathrm{Gr}(d,h)| \cong |\mathrm{Gr}(d,h)^{\lozenge}|.

It turns out that one can actually get a handle on \mathrm{Gr}(d,h)^{na} in a bunch of cases!  This grew out of some conversations with Jared Weinstein – back in April, Jared raised the question of understanding the inadmissible locus in a certain particular period domain for \mathrm{GL}_2 with non-minuscule Hodge numbers, and we managed to describe it completely in that case (see link below). Last night, though, I realized we hadn’t worked out any interesting examples in the minuscule (i.e. p-divisible group) setting! Here I want to record two such examples, hot off my blackboard, one simple and one delightfully bizarre.

Example 1. Take h=4, d=2 and H isoclinic. Then |\mathrm{Gr}(d,h)^a|^c is a single classical point, corresponding to the unique filtration on M(H) with Hodge numbers 0,0,1,1 which is not weakly admissible. So \mathrm{Gr}(d,h)^a = \mathrm{Gr}(d,h)^{wa} in this case.

Example 2. Take h=5, d=2 and H isoclinic$.  Now things are much stranger.  Are you ready?
Theorem. In this case, the locus \mathrm{Gr}^{na} is naturally isomorphic to the diamond (X \smallsetminus 0)^{\lozenge} / \underline{D^\times}, where X is an open perfectoid unit disk in one variable over \breve{\mathbf{Q}}_p and D=D_{1/3} is the division algebra over \mathbf{Q}_p with invariant 1/3, acting freely on X \smallsetminus 0 in a certain natural way. Precisely, the disk X arises as the universal cover of the connected p-divisible group of dimension 1 and height 15, and its natural D-action comes from the natural D_{1/15}-action on X via the map D_{1/3} \to D_{1/3} \otimes D_{-2/5} \simeq D_{-1/15} \simeq D_{1/15}^{op}.

This explicit description is actually equivariant for the D_{2/5}-actions on X and Gr. As far as diamonds go, (X \smallsetminus 0)^{\lozenge}/\underline{D^{\times}} is pretty high-carat: it’s spatial (roughly, its qcqs with lots of qcqs open subdiamonds), and its structure morphism to \mathrm{Spd}\,\breve{\mathbf{Q}}_p is separated, smooth, quasicompact, and partially proper in the appropriate senses. Smoothness, in particular, is meant in the sense of Definition 6.1 here (cf. also the discussion in Section 4.3 here). So even though this beast doesn’t have any points over any finite extension of \breve{\mathbf{Q}}_p, it’s still morally a diamondly version of a smooth projective curve!

The example Jared and I had originally worked out is recorded in section 5.5 here. The reader may wish to try adapting our argument from that situation to the cases mentioned above – this is a great exercise in actually using the classification of vector bundles on the Fargues-Fontaine curve in a hands-on calculation.

Anyway, here’s a picture of (X \smallsetminus 0)^{\lozenge} / \underline{D^{\times}}, with some other inadmissible loci in the background:




*All rigid spaces here and throughout the post are viewed as adic spaces: in the classical language, \mathrm{Gr}(d,h)^a does not generally correspond to an admissible open subset of \mathrm{Gr}(d,h), so one would be forced to say that there exists a rigid space \mathrm{Gr}(d,h)^a together with an etale monomorphism \mathrm{Gr}(d,h)^a \to \mathrm{Gr}(d,h). But in the adic world it really is a subspace.