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.

4 thoughts on “Several things”

  1. Maybe you should invent a new notion that would provide the same functionality to an arithmetic geometer as adic spaces that would not require valuation theory as background!

    Liked by 1 person

    1. I highly doubt this could be done – like I said, you have to eat your spinach first, and any theory of nonarchimedean analytic spaces with enough expressive power will inevitably come with some spinach. Adic spaces are great!

      Liked by 1 person

  2. If you dislike valuations, you are in good company: there is a letter in the Grothendieck-Serre correspondence (October 19, 1961) where Grothendieck rants about valuations (how they are useless, misleading and should be banned to the obscur bottom of “things not to read”). Serre, in his response (October 26, 1961), objects to them being useless, mentioning how useful they are to …p-adic people (Tate, Dwork). Cheers!

    Liked by 1 person

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