Thoughts from Oberwolfach, Part I

Just back from a week-long workshop at Oberwolfach on “Nonarchimedean geometry and applications”. Some random comments:

  • During the workshop, Kiran Kedlaya started a nonarchimedean Scottish book, to record open problems in the foundations of nonarchimedean geometry, especially in the theory of adic and perfectoid spaces. (See here if you don’t understand the name.) There’s no short supply of problems!
  • On a related note, Kiran convinced me (against my will) that given some Huber pair (A,A^+), the question of whether or not the structure presheaf on X=\mathrm{Spa}(A,A^+) is a sheaf depends a priori on the choice of A^+! The issue is that for two different choices A^+ \subseteq A'^+ with associated X' \subseteq X, rational subsets match up but coverings by rational subsets might not. Depressing.
    On the other hand, every practical criterion for sheafyness (e.g. locally Noetherian, f.g. as an algebra over a Noetherian ring of definition, perfectoid, stably uniform, etc.) is independent of A^+.
  • Werner Lutkebohmert is very nice.
  • There’s a natural notion of a “perfectoid Deligne-Mumford stack”. To make this precise, let K/\mathbf{Q}_p be a nonarchimedean field.  Then (provisionally) a perfectoid DM stack over K is a stack in groupoids \mathcal{X} on \mathrm{Perf}^{\mathrm{proet}}_{/ \mathrm{Spd}\,K} whose diagonal \mathcal{X} \to \mathcal{X} \times_{\mathrm{Spd}\,K} \mathcal{X} is representable in diamonds and which admits a surjective and pro-etale 1-morphism \mathrm{Perf}^{\mathrm{proet}}_{/U}\to\mathcal{X} for some U\in\mathrm{Perf}_{/ \mathrm{Spd}\,K}.  (As is customary, we’ll abbreviate this last thing to “a morphism U \to \mathcal{X}“.)
    One can make examples of these things by analytifying classical Deligne-Mumford stacks over \mathrm{Spec}\,K. Not very surprising (although it requires proof).  But here’s a much weirder example: Take K=\widehat{F^{nr}} for some finite F/\mathbf{Q}_p, and let D be the division algebra over F with invariant 1/n.  Then there’s a natural action of D^{\times} on \mathbf{P}^{n-1}_{K}, and the groupoid [\mathbf{P}^{n-1,\lozenge}_{K}/D^{\times}] is a perfectoid DM stack over K. Roughly speaking, the idea is that the infinite-level Lubin-Tate space gives a pro-etale and D^{\times}-equivariant \mathrm{GL}_n(F)-torsor \mathcal{M}_{\mathrm{LT},\infty} \to \mathbf{P}^{n-1}_{K}, so then quotienting we get [\mathcal{M}_{\mathrm{LT},\infty}^{\lozenge}/D^{\times}]\to [\mathbf{P}^{n-1,\lozenge}_{K}/D^{\times}] surjective and pro-etale. But \mathcal{M}_{\mathrm{LT},\infty}\cong\mathcal{M}_{\mathrm{Dr},\infty} matches up with Drinfeld space at infinite level, and then [\mathcal{M}_{\mathrm{LT},\infty}^{\lozenge}/D^{\times}]\cong[\mathcal{M}_{\mathrm{Dr},\infty}^{\lozenge}/D^{\times}]\cong \Omega^{n-1,\lozenge} is the diamond associated with the Drinfeld upper half-space. This gives the required U.  Anyway, I’ll try to write this up more carefully in the next version of this thing.
    Nb. The notion of a perfectoid DM stack (and even of a perfectoid algebraic stack) will play a crucial role in Fargues’s geometrization of the local Langlands correspondence, so there’s ample motivation for considering these objects.
  • Antoine Ducros takes very good photos.
  • Prize for Best Dressed: Wiesława Nizioł, Fumiharu Kato.
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