Fix a complete nonarchimedean field 
Definition. A surjection 
Being distinguished imposes some obvious conditions on
1)
Since any residue norm takes values in 
2) 
If 
All of this can be found in section 6.4.3 of BGR.
Question 1. If 
This should be easy if it’s true. I didn’t think much about it.
Now suppose 
Loosely following Bosch, let us say an affinoid subdomain 
1) all 
2) for each 
This is a very clean kind of affinoid cover: we can immediately build a formal model for 
Question 2. For 
be an open immersion of an affine scheme into a smooth projective variety. Is the complement 
an ample divisor on 
be an open immersion of affine schemes. Is 
the nonvanishing locus of a section of an ample line bundle on 
?
, there is an equivalence of categories from 
-representations of 
towards semistable Higgs bundles on 
setting). Of course on the spectral side, one expects to see some category of quasicoherent or ind-coherent sheaves on the special fiber of the Emerton-Gee stack for 
. On the automorphic side, one should have some category of mod-p sheaves on 
, and the correct category should fall out of the general formalism developed by Mann in his thesis. One tantalizing fact, sketched out during some of these conversations, is that 
-cohomologically smooth (in a precise sense), not of dimension 0 as in the 
case, but of dimension equal to the dimension of 
as a 
and 
appearing in 
used by FS, which are 
-cohomologically smooth for all primes 
. Likewise, their “strict Henselian” property should fail badly. New ideas are very much required!
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 
between Artin v-stacks. For maps which are actually “nice”, and in particular non-stacky, these functors were constructed by Scholze, but extending them to the setting of stacky maps is not formal. Already in the analogous setting of schemes versus Artin stacks, these functors for Artin stacks were only constructed recently by 
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 
functors in the 
setting (for 
a locally closed immersion), one can define functors 
in the 
is the inclusion of any Harder-Narasimhan stratum into 
as the composition 
. Here 
is the open substack of bundles which are “more semistable” than 
. It will also be convenient to write 
. Note that 
is an open immersion, so 
clearly preserves 
as usual.
and its punctured version 
. Recall that these charts also come with compatible maps ![q_b: \mathcal{M}_b \to [\ast / G_b(E)]](https://s0.wp.com/latex.php?latex=q_b%3A+%5Cmathcal%7BM%7D_b+%5Cto+%5B%5Cast+%2F+G_b%28E%29%5D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
and ![q_b^{\circ}: \mathcal{M}_{b}^{\circ} \to [\ast / G_b(E)]](https://s0.wp.com/latex.php?latex=q_b%5E%7B%5Ccirc%7D%3A+%5Cmathcal%7BM%7D_%7Bb%7D%5E%7B%5Ccirc%7D+%5Cto+%5B%5Cast+%2F+G_b%28E%29%5D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
. Then for any ![A \in D_{lis}(\mathrm{Bun}_{G}^{b},\Lambda) \cong D_{lis}([\ast / G_b(E)],\Lambda)](https://s0.wp.com/latex.php?latex=A+%5Cin+D_%7Blis%7D%28%5Cmathrm%7BBun%7D_%7BG%7D%5E%7Bb%7D%2C%5CLambda%29+%5Ccong+D_%7Blis%7D%28%5B%5Cast+%2F+G_b%28E%29%5D%2C%5CLambda%29&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
, the correct definition turns out to be
.
for cohomologically smooth maps 
and 
are cohomologically smooth – one of the hardest theorems in the paper! – the above construction preserves 
is just 
clearly vanishes.
turned out to be surprisingly subtle, cf. Theorem 3.39. Note that 
, and the latter carries an obvious t-structure. Nevertheless, we weren’t able to settle the question of whether these t-structures are compatible:
on 
should actually also be OK.) The first basic point to make is that for any rigid space 
, any object 
is ULA for the structure map 
. Sketch: The claim is local on 
is constant. By an argument with resolution of singularities, we now reduce further to the case where 
-coefficients, or with general 
coefficients (but then only for objects of “finite tor-dimension”).
of proper rigid spaces over an algebraically closed field, and a cohomological correspondence 
on some 
. Then the usual recipe to define local terms applies, and the expected equality 
holds true. (Note that 
is a perfect 
are just as mysterious as in the case of schemes.
be a proper map of characteristic zero rigid spaces, and let 
over which 
is Hodge-Tate, or the even weaker statement that it has integral Hodge-Tate-Sen weights.
, ii) non-algebraizable deformations of K3 surfaces over the residue field of 
, does there exist a proper 
-dimensional rigid space with no non-constant meromorphic functions, and admitting a formal model whose special fiber has components of general type? Can we find examples of such spaces with arbitrarily large dimension dimension which don’t come from lower-dimensional examples by simple operations (products, quotients by finite groups, etc.)? Same question but with “no non-constant meromorphic functions” replaced by the weaker requirement that 
is small compared to 
.
and then construct explicit charts for closed Schubert cells, using moduli of local shtukas at infinite level. You can read Bence’s outstanding thesis 
. Let 
where 
is a vector bundle of rank 
is an injection which is an isomorphism outside the closed Cartier divisor at infinity.
with a Weil descent datum to 
be the division algebra over 
, and let 
be an irreducible representation of 
.](https://s0.wp.com/latex.php?latex=R%5CGamma_c%28X_%7B%5Cmathbf%7BC%7D_p%7D%2C%5Coverline%7B%5Cmathbf%7BQ%7D_%5Cell%7D%29%5Cotimes_%7BD%5E%5Ctimes%7D+%5Ctau+%5Ccong+%5Cvarphi_%7B%5Ctau%7D%5B1-2n%5D%28%5Ctfrac%7B1-2n%7D%7B2%7D%29&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
if 
if 
denotes the Langlands parameter of 
is naturally a 
-torsor over 
totallydisconnected 