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: Letand
be separated morphisms of locally spatial diamonds such that
is surjective and universally open, and
is compactifiable (in the sense of Definition 22.2 here). Is
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
-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.
in etale cohomology for certain stacky maps of Artin v-stacks, by adapting some machinery of Liu-Zheng which they built to solve the analogous problem in the setting of Artin stacks. Since the above-mentioned papers of Hamann and Koshikawa both depend directly on HKW, and thus indirectly on GHW, I’ve felt some increased pressure recently* to actually produce this paper!
-categorical constructions. Actually, this is the most intense and frustrating project I’ve ever worked on. It will be good to finish it.
-affine Grassmannian, Theorem 5.1.3, now has a completely different and correct (!) proof, via a degeneration to a similar calculation on the Witt vector affine Grassmannian. This degeneration argument relies on the observation that the local terms appearing in the Lefschetz-Verdier trace formula are compatible with any base change, Proposition 5.3.1; this seems to be a new observation even for schemes. (My first proof of Proposition 5.3.1 involved checking the commutativity of 500 diagrams, but eventually I realized that it follows immediately from the Lu-Zheng formalism!) Once we’ve degenerated to the Witt vector affine Grassmannian, we make an intriguing global-to-local argument using the trace formula and known properties of the weight functors in geometric Satake, and relying crucially on a 
-lattice” have now been eliminated. This is far from formal, since we definitely don’t have a 6-functor formalism with 
-coefficients at hand. Instead, the idea is to reduce “by a continuity argument” to the case where things do admit lattices. The key ingredient here is Theorem 6.5.4, formerly a conjecture of Taylor, which should have lots of other applications.
in etale cohomology for “smooth-locally nice” maps 
restricted to elliptic elements of 
? What’s so special about elliptic elements in this context?
and 
, so we’re in the Lubin-Tate/Drinfeld setting with 
the units in the quaternion algebra over 
. Let 
be the trivial representation of 
. Then 
as a virtual representation of 
is a principal series representation, hence non-elliptic, so the virtual character of 
on elliptic elements of 
. This matches perfectly with the fact that any elliptic 
has two fixed points in 
, both contained in 
, and the “naive” local terms of the relevant sheaf ![j_!\mathbf{Z}_{\ell}[1]](https://s0.wp.com/latex.php?latex=j_%21%5Cmathbf%7BZ%7D_%7B%5Cell%7D%5B1%5D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
at both these points are 
. Here 
is the evident open immersion.
is regular semisimple and nonelliptic, then it’s conjugate to some 
with 
. In this case there are still two fixed points, but they both lie in the “boundary” 
. Since 
. This character value can be computed explicitly by van Dijk’s formula, and turns out to be 
. So this slightly strange expression needs to emerge from the sum of these two local terms. 
with 
the Steinberg parameter. Then 
. So in this case 
is, up to sign, a reducible principal series representation.
, and indeed Fargues-Scholze handles these two situations uniformly. However, the literature on local Langlands and harmonic analysis for reductive groups over positive characteristic local fields case is pretty thin. This is the main reason we stuck to the mixed characteristic case. For more on the state of the literature, take a look at section 1.6 in 
for non-basic b?
is non-basic, or 
. This is a soft form of the Harris-Viehmann conjecture. The details of this argument will be given elsewhere (in the sense of Deligne-Mumford).
and Hecke operator formalism of [FS21] with general coefficient rings.
. It’s naturally of interest to wonder how everything depends on this choice. Imai has formulated a 
” results in the etale cohomology of diamonds are probably very hard.
might look like (no prizes for guessing who drew this):
be some p-divisible group of dimension d and height h, and let 
be the rigid generic fiber (over 
) of the associated Rapoport-Zink space. This comes with its Grothendieck-Messing period map 
, where 
is the rigid analytic Grassmannian paramatrizing rank d quotients of the (covariant) rational Dieudonne module 
. Note that 
is etale and partially proper (i.e. without boundary in Berkovich’s sense), and so the image of 
. 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 
) and 
is connected, we’re in the much-studied Lubin-Tate situation. Here, Gross and Hopkins famously proved that 
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 
, cf. Rapoport’s appendix to Scholze’s paper on the Lubin-Tate tower.
, i.e. when 
is isomorphic to rigid analytic affine space of the appropriate dimension, and can be identified with the open Bruhat cell inside 
, 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 
is etale and bijective; this is the terminology used e.g. in Rapoport-Zink’s book.
is irreducible as a 
-module), 
, so the weakly admissible locus tells you zilch about the admissible locus in this situation (and they really are different for any 
).
is open and specializing, so its complement is closed and generalizing. Now, according to a very general theorem of Scholze, namely Theorem 2.42 
is any diamond and 
is any locally closed generalizing subset, there is a functorially associated subdiamond 
with 
inside 
. More colloquially, one can “diamondize” any locally closed generalizing subset of 
for a scheme 
comes from a unique (reduced) subscheme of 
is the subdiamond of 
obtained by diamondizing the topological complement of the admissible locus, i.e. by diamondizing the closed generalizing subset 
.
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.
is a single classical point, corresponding to the unique filtration on 
which is not weakly admissible. So 
in this case.
is naturally isomorphic to the diamond 
, where 
and 
is the division algebra over 
in a certain natural way. Precisely, the disk 
-action comes from the natural 
-action on 
.
-actions on 
. As far as diamonds go, 
is pretty high-carat: it’s spatial (roughly, its qcqs with lots of qcqs open subdiamonds), and its structure morphism to 
is separated, smooth, quasicompact, and partially proper in the appropriate senses. Smoothness, in particular, is meant in the sense of Definition 6.1 
, with some other inadmissible loci in the background:
. But in the adic world it really is a subspace.
totallydisconnected 