the March of progress

Several things to report on:

  • The (hopefully) final version of HKW is done and dusted! The most significant change from the previous version is that GHW is finally done, so we can finally refer to precise results from that paper. For a quick summary of the main results from GHW, take a look at section 4.1 of HKW. The key new vocab is decent v-stacks and fine morphisms between them. Writing GHW was not good for my health (take a look at the acknowledgments…), and reading it might not be so good for yours. But if you really want to look at GHW, one thing you could do to get oriented is read Definitions 1.1 & 1.3, Theorem 1.4, and everything in section 4.1.

    Bonus Question: Let g:X \to Y and f:Y \to Z be separated morphisms of locally spatial diamonds such that g is surjective and universally open, and f \circ g is compactifiable (in the sense of Definition 22.2 here). Is f compactifiable? If the answer to this question is yes, then the nonsense about “strict” surjectivity in GHW could be eliminated (although it is harmless in applications, thank God). If you can settle this, or some slight weakening of it, please let me know!

    Note that the numbering in HKW has changed slightly in comparison to the previous version, which I wrote about here, so the numbering in that post refers to v3 of the arxiv posting.
  • In a previous post, I mentioned a bunch of roughly equivalent open problems about smooth representations of p-adic groups with coefficients in general \mathbf{Z}[1/p]-algebras, which I learned about from Jean-François Dat. But now these problems have been more or less all solved, in a beautiful and shockingly short paper by Dat-Helm-Kurinczuk-Moss. It is quite curious that their results, which are statements in pure representation theory, depend in a crucial way on the Fargues-Scholze machinery. For more information, I can’t to any better than suggesting that you simply read their paper.

Several things

A rant about valuations
If you look at any detailed introductory text on adic spaces (e.g. the notes of Conrad, Morel, Wedhorn, etc.) you’ll find lots and lots and lots of preliminary material on valuation theory. On the one hand, this isn’t so crazy, since adic spaces are built from valuation spectra, and you have to eat your spinach before you get to have cake. On the other hand, I think this is pretty unfortunate, since valuation theory is incredibly boring and dry, and (more importantly) when you actually work with analytic adic spaces in real life, you never need to worry about most of this material. How often have I worried about horizontal specializations versus vertical specializations, etc.? Essentially never.

Open problems
Suppose you want to study representation theory of p-adic reductive groups with coefficients in some Noetherian ring R with p \in R^\times. You might be surprised to learn that the following basic results are all unknown in general (as Jean-Francois Dat kindly pointed out to me), even when R is a DVR:
-parabolic induction preserves finitely generated objects,
-Jacquet modules preserve admissible objects,
-products of cuspidal objects are cuspidal,
-the category is Noetherian,
-second adjointness holds.
I believe the state of the art here is a paper of Dat from 2009, which explains the interrelations between these problems and solves them for many groups. Does anyone have in mind a strategy to solve these problems completely? I would love to know.