## Vanishing for O+/p-cohomology of Stein spaces

OK I’m too lazy to convert this into a proper blog post, but Proposition 0.4 here is something I needed recently, which others might find useful.  Lemma 0.2 might also come in handy.  Please ask in the comments if you’d like more details!

Update (Sept. 29). James Newton pointed out that it’s not actually clear how to fill in the details in the “proof” of Lemma 0.1.i. Therefore, the results in Lemmas 0.1 and 0.2 of the document linked above are currently unproved in the generality stated there. Fortunately, I was able to recover some weaker versions of Lemmas 0.1 and 0.2 which still suffice for the intended application to cohomology of Stein spaces. The corrected writeup is available here.

## Zariski closed immersions

In p-adic geometry, what should it mean for a morphism to be a Zariski-closed immersion? For locally Noetherian adic spaces, the usual notion of a closed immersion of locally ringed spaces works just fine. For general analytic adic spaces, though, one quickly runs into annoying foundational issues. The issue is roughly as follows. Let $X=\mathrm{Spa}(A,A^+)$ be an (analytic) affinoid adic space. We can certainly define a reasonable notion of Zariski-closed subset, just by following our nose: a subset $Z \subset |X|$ should be Zariski-closed if there is an ideal $I \subset A$ such that $x \in |Z|$ iff $|f|_x = 0\,\forall f \in I$. These are exactly the subsets obtained by pulling back closed subsets of $\mathrm{Spec}(A)$ along the natural map $|\mathrm{Spa}(A,A^+)| \to |\mathrm{Spec}(A)|$. The problem, however, is that such a $Z$ need not come from an actual closed immersion of an affinoid adic space into $X$, because the quotient $A/I$ could just be some junky non-sheafy ring, and maybe there’s no canonical tweak (like replacing $I$ by its closure, or replacing $A/I$ by its uniform completion, or…) which will make it sheafy. And even if we can tweak $A/I$ to make it sheafy, how do we know that $A \to A/I$ is still surjective after going to some rational subset $U \subset X$? You get the picture.

Perhaps surprisingly, the situation for affinoid perfectoid spaces is a lot better. In particular, if $(A,A^+)$ is a perfectoid Tate-Huber pair, there are canonical bijections (satisfying some obvious compatibilities) between
1) closed subsets of $\mathrm{Spec}(A)$,
2) Zariski-closed subsets of $X=\mathrm{Spa}(A,A^+)$,
3) (isomorphism classes of) maps of Tate-Huber pairs $(A,A^+) \to (B,B^+)$ where $B$ is a perfectoid Tate ring, $A \to B$ is surjective, and $B^+$ is the integral closure of the image of $A^+$ in $B$.

We’ve already discussed the bijection 1) <–> 2). For 3) –> 1) or 2), just send $q:A \twoheadrightarrow B$ to the closed subset cut out by the ideal $\ker q$.  The miracle is the association 2) –> 3), which holds by an amazing theorem of Bhatt: if $I$ is a closed ideal in a perfectoid Tate ring $A$, then the uniform completion $B$ of $A/I$ is perfectoid and the natural map $A \to B$ is surjective, cf. Theorem 2.9.12 in Kedlaya’s notes here. Moreover, the map $A \to B$ remains surjective after rational localization on $A$. In particular, if $Z \subset X$ is a Zariski-closed subset, then 2) –> 3) gives an honest closed immersion $\mathrm{Spa}(B,B^+) \to \mathrm{Spa}(A,A^+)$ of locally ringed spaces, and $|\mathrm{Spa}(B,B^+)|$ maps homeomorphically onto $Z$.

The point of all this is that Zariski-closed immersions of affinoid perfectoid spaces behave as well as one could ever dream (with one caveat, which I’ll get to later). The following definition then suggests itself.

Definition. A map of small v-stacks $X \to Y$ is a Zariski-closed immersion if for any affinoid perfectoid space $W$ with a map $W \to Y$, the base change $X \times_{Y} W \to W$ is a Zariski-closed immersion of affinoid perfectoid spaces.

Now of course we’re free to make any definition we want in mathematics, but if it doesn’t capture some essential idea or experimentally observed phenomenon, then who cares? Let me now give some evidence that this definition passes this test.

Example 0. The property of being a Zariski-closed immersion is preserved under composition and base change. If $X \to Y$ is a Zariski-closed immersion and $Y$ is (a small v-sheaf, a diamond, a locally spatial diamond, qc or qs or separated or partially proper over a base $S$), then so is $X$.

Example 1. Let $f: X \to Y$ be a closed immersion of locally Noetherian adic spaces. If $Y$ is affinoid (so $X$ is too), then the map of diamonds $f^{\lozenge} : X^{\lozenge} \to Y^{\lozenge}$ is a Zariski-closed immersion. This is easy.

Example 2. Let $f: X \to Y$ be a closed immersion of locally Noetherian adic spaces again, but now assume that $f$ is the analytification of a closed immersion of quasiprojective varieties.  Then $f^{\lozenge}: X^{\lozenge} \to Y^{\lozenge}$ is a Zariski-closed immersion.  For this, we can use the assumption on $f$ to choose a vector bundle $\mathcal{E}$ on $Y$ together with a surjection $\mathcal{E} \twoheadrightarrow \mathcal{I}_{X} \subset \mathcal{O}_Y$. Then for any map $g: W \to Y$ from an affinoid perfectoid, the pullback $g^{\ast}\mathcal{E}$ (in the usual sense of ringed spaces) is a vector bundle on $W$, hence generated by finitely many global sections $e_1,\dots,e_n$ by Kedlaya-Liu. The images of $e_1,\dots,e_n$ along the natural map $(g^{\ast}\mathcal{E})(W) \to \mathcal{O}_{W}(W)$ generate an ideal, and the associated closed immersion of affinoid perfectoids $V \to W$ represents the fiber product $X^{\lozenge} \times_{Y^{\lozenge}} W$. (Hat tip to PS for suggesting this vector bundle trick.)

Example 3. Let $X^{\ast}$ be a minimally compactified Hodge-type Shimura variety with infinite level at $p$. Then the boundary $Z \to X^{\ast}$ is a Zariski-closed immersion, and so is the diagonal $X^{\ast} \to X^{\ast} \times X^{\ast}$. (These both reduce to the previous example, using a small limit argument in the second case.) In particular, if $U,V \subset X^{\ast}$ are any open affinoid perfectoid subsets, then $U \cap V$ is also affinoid perfectoid. This small observation plays a non-negligible role in my forthcoming paper with Christian Johansson, where (among other things) we prove that any minimally compactified Shimura variety of pre-abelian type with infinite level at $p$ is perfectoid.

Example 4. Fix a perfectoid base field $K$ of characteristic zero. Then the inclusions $\mathrm{Fil}^n \mathrm{B}_{\mathrm{dR}}^{+} \subset \mathrm{B}_{\mathrm{dR}} \subset \mathrm{B}_{\mathrm{dR}}$ are Zariski-closed immersions of (ind-)diamonds over $\mathrm{Spd} K$. This can be proved by induction on $n$, and the base case reduces to the fact that the inclusion $\mathrm{Fil}^1 \mathrm{B}_{\mathrm{dR}}^{+} \subset \mathrm{B}_{\mathrm{dR}}^{+}$ is the pullback of $\{ 0 \} \to \mathbf{A}^{1}_{K}$ along $\theta$. (To make the induction work, you need to pick an element $\xi \in \mathrm{B_{dR}}^+(K)$ generating $\ker \theta$.)

Example 5. Fix a complete algebraically closed extension $C / \mathbf{Q}_p$. Fix a reductive group $G / \mathbf{Q}_p$ and a geometric conjugacy class of $G$-valued cocharacters $\mu$. Then $\mathrm{Gr}^{\mathrm{B_{dR}}}_{G, \leq \mu, C} \to \mathrm{Gr}^{\mathrm{B_{dR}}}_{G, C}$ is a Zariski-closed immersion. Also, if $\nu \leq \mu$, then $\mathrm{Gr}^{\mathrm{B_{dR}}}_{G, \leq \nu, C} \to \mathrm{Gr}^{\mathrm{B_{dR}}}_{G, \leq \mu, C}$ is a Zariski-closed immersion. These claims can be reduced to the case $G = \mathrm{GL}_n$, which in turn reduces to Example 4 by some trickery.

Example 6. Fix a complete algebraically closed nonarchimedean field $C$ of residue characteristic $p$, and let $\mathcal{E} \to \mathcal{F}$ be any injective map of coherent sheaves on the Fargues-Fontaine curve $X_C$. Then the associated map of Banach-Colmez spaces $\mathbb{V}(\mathcal{E}) \to \mathbb{V}(\mathcal{F})$ is a Zariski-closed immersion. This can also be reduced to Example 4.

Let me end with some caveats. First of all, I wasn’t able to prove that if $G \to H$ is a closed immersion of reductive groups, the induced map $\mathrm{Gr}^{\mathrm{B_{dR}}}_{G, C} \to \mathrm{Gr}^{\mathrm{B_{dR}}}_{H, C}$ is a Zariski-closed immersion, although it is surely true. The problem here is (roughly) that an $H$-torsor over some affinoid perfectoid $X$ can only be reduced to a $G$-torsor locally in the analytic topology on $X$, and we then run into the following open question:

Question. Is the property of being Zariski-closed local for the analytic topology? More precisely, if $X$ is affinoid perfectoid with a covering by rational subsets $U_i$, and $Z$ is a closed subset such that $Z \cap U_i$ is Zariski-closed in $U_i$ for all $i$, is $Z$ Zariski-closed?

There are also naturally occurring closed things which probably aren’t Zariski-closed immersions. For instance, I don’t think the map of Banach-Colmez spaces $0 \to H^1(\mathcal{O}(-1)) = \mathbf{A}^{1,\lozenge}_{C} / \underline{\mathbf{Q}_p}$ is a Zariski-closed immersion, because then pulling back would imply that $\underline{\mathbf{Q}_p} \to \mathbf{A}^{1,\lozenge}_{C}$ is a Zariski-closed immersion, which seems extremely unlikely to me. (But I didn’t manage to disprove it! Actually, can one give an explicit example of an affinoid perfectoid $X/C$ and a closed subset $S \subset X$ such that $C$ maps isomorphically to the completed residue field at every point in $S$ and such that $S$ is NOT Zariski-closed? Surely such examples exist.) I also don’t think (closures of) Newton strata in flag varieties are Zariski-closed immersions – they are just too weird.

I also wasn’t able to settle the following compatibility (but admittedly I didn’t try very hard).

Question. Let $f: X \to Y$ be a monomorphism of locally Noetherian adic spaces. If $f^{\lozenge}$ is a Zariski-closed immersion, is $f$ actually a closed immersion?

Happy new year!

## spectral spaces; snark

(Update 12/24: Unfortunately the argument below doesn’t work as written. I overlooked the following detail in my “proof” that $X^{wl} \to X$ is open: if $S \to S'$ is a surjective continuous map of finite $T_0$ spaces, it’s not necessarily true that $S^Z \to S'^Z$ is surjective. For instance, if $S'= \{ \eta \rightsquigarrow s \}$ is the spectrum of a DVR and $S=\{ \eta, s \}$ is $S'$ equipped with the discrete topology, then $S'^Z$ has three points but $S=S^Z$ has two.

For the argument below to work, it would be enough to know that for any open subset $V \subset X^{wl}$, its image in $X$ contains an open subset. Is this true?

In any case, the Corollary is still true, although by a totally different argument.)

*****

I spent about six hours yesterday and today proving the following thing.

Lemma. Let $X$ be a spectral space, and let $Z$ be a closed generalizing nowhere-dense subset of $X$. Then $Z$ is nowhere-dense for the constructible topology on $X$, i.e. $Z$ doesn’t contain any nonempty constructible subset of $X$.

This has the following concrete consequence, which is what I really needed.

Corollary. Let $(A,A^+)$ be some Tate-Huber pair with adic spectrum $X=\mathrm{Spa}(A,A^+)$, and let $Z \subset X$ be a Zariski-closed nowhere-dense subset. Suppose $U_1$ and $U_2$ are quasicompact open subsets of $X$ such that $U_1 \cap (X - Z) = U_2 \cap (X - Z)$. Then $U_1 = U_2$.

Proof of Corollary. We need to check that $V= (U_1 \cup U_2 ) - (U_1 \cap U_2)$ is empty. But $V$ is a constructible subset of $X$ contained in $Z$, so this is immediate from the lemma.

Amusingly, even though this corollary is pretty down-to-earth, I only managed to prove it by proving the lemma, and I only managed to prove the lemma by exploiting the structure of the w-localization $X^{wl}$ of $X$. Is there a more direct approach? Am I missing something obvious?

(Sketch of actual argument: the profinite set of closed points $X^{wl}_{c}$ maps homeomorphically onto $X$ equipped with the constructible topology, so if $V \subset X$ is constructible it is clopen when viewed as a subset of $X^{wl}_{c}$. The key point is then to check that $Z$ is nowhere-dense when viewed as a subset of $X^{wl}_c$. This can be done, using that the natural surjection $t: X^{wl} \to X$ is open and that $t^{-1}(Z)$ (which is then closed, generalizing and nowhere-dense in $X^{wl}$, the last point by openness of $t$) is the preimage of its image in $\pi_0(X^{wl}) \cong X^{wl}_c$.

The openness of $t$ doesn’t seem to be stated in the literature, but it can be deduced from the proof of Lemma 2.1.10 in Bhatt-Scholze, using the fact that it’s obviously true for finite $T_0$ spaces.)

You may have noticed that RIMS is hosting a series of four workshops next year under the umbrella of a “RIMS Research Project” entitled Expanding Horizons of Inter-universal Teichmuller Theory.  The first of the workshops looks pretty reasonable, the other three not so much.  In case you’re wondering (as I did) how much money RIMS is ponying up for this, it seems to be capped at 5 million yen, or about \$41k (according to e.g. this document). This doesn’t seem like very much money to support four workshops; I guess some funding is also coming from that infamous EPSRC grant.

Anyway, when you’re inside a black hole, your horizons might seem quite expansive indeed, but I doubt you’ll have much luck convincing others to join you.

## p-adic Kahler manifolds

In complex geometry, the most interesting class of complex manifolds is probably the Kahler class. In the non-archimedean world, say over a fixed p-adic base field $K$, the analogue of a compact complex manifold is a smooth proper rigid analytic space. In some ways, these are already surprisingly “close” to being Kahler – in particular, the Hodge-de Rham spectral sequence of such a space always degenerates at $E_1$. However, Hodge symmetry can definitely fail. A standard example is the non-archimedean Hopf surface $X = \mathbf{A}^2_{K} \smallsetminus \{ (0,0) \} / p^{\mathbf{Z}}$ where $p^n$ acts through diagonal multiplication. By a fun direct calculation, one checks that $H^0(X,\Omega^1_X)=0$ and $H^1(X,\mathcal{O}_X) = K$, so Hodge symmetry fails in degree one.

We now see a natural question: is there is some non-archimedean analogue of the Kahler condition which restores Hodge symmetry? Two years ago, Shizhang Li hit upon the following candiate condition:

A smooth proper rigid space $X$ satisfies (*) if it admits a formal model $\mathfrak{X}$ over $\mathcal{O}_K$ whose special fiber is projective (as opposed to merely proper).

Using fantastic ideas due to Shizhang, we managed to prove the following suggestive result.

Theorem. Let $X$ be a smooth proper rigid space satisfying (*). Then $h^{1,0}(X) = h^{0,1}(X)$.

Of course, one can then guess that (*) implies Hodge symmetry in all degrees. This speculation seems to have caught the imagination of others in the field, but until recently I personally regarded it as not much more than wishful thinking. However, my perspective completely changed a month ago, when I learned from Shizhang that, according to Robert Friedman, the archimedean analogue of “(*) implies Hodge symmetry” is a theorem! More precisely, we have the following result:

Theorem. Let $D$ be the complex disk, with $D^\times =D \smallsetminus \{0 \}$ the punctured disk. Let $f:Y \to D$ be a proper map of complex analytic spaces. Suppose that $f^{-1}(D^\times) \to D^\times$ is a submersion, and that the central fiber $Y_0=f^{-1}(0)$ is the analytification of a projective (and not necessarily smooth) algebraic variety. Then for all $t \in D^\times$ with $|t| \ll 1$, the fiber $Y_t$ satisfies Hodge symmetry and Hodge-de Rham degeneration.

Of course, the analogy is that $\mathfrak{X} \to \mathrm{Spf} \mathcal{O}_K$ is analogous to $Y \to D$, and $X$ is analogous to the “nearby” fibers $Y_t$ with $0<|t| \ll 1$.

The proof of this theorem uses the full power of mixed Hodge theory. In fact the claim about Hodge-de Rham degeneration is exactly Corollary 11.24 in the book of Peters-Steenbrink. Hodge symmetry is even more subtle, and the argument for this doesn’t seem to be written down anywhere; Friedman explained it to Shizhang, who explained it to me, but the details entailed such a horrible explosion of gradings, filtrations, and multi-indices that I can’t hope to reproduce it here.

Anyway, I’m now completely convinced that Shizhang’s condition (*) implies Hodge symmetry in all degrees, and that this is truly the “right” p-adic analogue of the Kahler condition.

## sheafiness; perversity

$\bullet$ One of the main annoyances in the theory of adic spaces is that, for a given Huber pair $(A,A^+)$, the structure presheaf on $\mathrm{Spa}(A,A^{+})$ is not always a sheaf. One usually remedies this by restricting attention to various classes of Huber rings, e.g. strongly Noetherian Tate rings, perfectoid Tate rings, Noetherian adic rings, etc. However, the following class of rings doesn’t seem to be addressed in the literature:

Definition. Let $A$ be an adic ring with finitely generated ideal of definition $I \subset A$. We say $A$ is strongly Noetherian outside $I$ if, for all $n \geq 0$, the scheme $\mathrm{Spec}\,A\left\langle X_1,\dots,X_n \right\rangle \smallsetminus V(IA\left\langle X_1,\dots,X_n \right\rangle )$ is Noetherian.

Here $A\left\langle X_1,\dots,X_n \right\rangle$ denotes the usual ring of restricted power series. Note that if $A$ is a Tate ring and $(A_0,(\varpi))$ is any couple of definition, then $A$ is strongly Noetherian if and only if $A_0$ is strongly Noetherian outside $(\varpi)$. I should also point out that the condition of being strongly Noetherian outside $I$ is already considered in a very interesting paper of Fujiwara-Gabber-Kato; they use the terminology “topologically universally rigid-Noetherian”, but I prefer my terminology on account of the previous sentence. Anyway, the following conjecture seems reasonable:

Conjecture. If $A$ is strongly Noetherian outside $I$, the structure presheaf on $\mathrm{Spa}(A,A)$ is a sheaf.

This implies that any strongly Noetherian Tate ring is sheafy (which of course is already known), but it also implies e.g. that if $A$ is topologically finitely presented over $\mathcal{O}_K$ for some nonarchimedean field $K$, then $A$ is sheafy. Sheafiness in the latter situation is known when $K$ is discretely valued, but to the best of my knowledge it’s open for general $K$.

I’m sure this conjecture is within reach, and maybe it’s an easy exercise for experts on sheafiness (*cough* Kiran *cough*). Note that FGK already proved some interesting consequences of this definition, which are probably relevant to proving this conjecture. Precisely, they show that if $A$ is strongly Noetherian outside $I$, then:
1. The $I$-power-torsion submodule of any finitely generated $A$-module is killed by a power of $I$.
2. If $N \subset M$ is any inclusion of $A$-modules, with $M$ finitely generated, then the subspace topology on $N$ induced by the $I$-adic topology on $M$ coincides with the $I$-adic topology on $N$.

$\bullet$ Let $j:U \to X$ be some immersion of varieties over a separably closed field. Everyone knows that the intermediate extension functor $j_{!\ast}$ on perverse sheaves (say with coefficients in $\Lambda = \mathbf{Q}_{\ell}$) is pretty great: it’s totally canonical, it commutes with Verdier duality, it preserves irreducibility, it preserves monic and epic maps of perverse sheaves, etc.

Recently I noticed that if $f: Z \to X$ is any map of varieties, with $X$ smooth and $Z$ irreducible, there’s still a natural functor $f^{\ast !}:\mathrm{Perv}(X) \to \mathrm{Perv}(Z)$ which commutes with Verdier duality. To define this functor, note that for any $f$ and any $\mathcal{F} \in D^b_c(X)$, there is a natural map $f^\ast \mathcal{F} \otimes Rf^! \Lambda \to Rf^! \mathcal{F}$, obtained by adjunction from the composite map $Rf_!( f^\ast \mathcal{F} \otimes Rf^! \Lambda) \cong \mathcal{F} \otimes Rf_! Rf^! \Lambda \to \mathcal{F}$ (the first isomorphism here is the projection formula). Since $X$ is smooth, the dualizing complex $\omega_X$ is just $\Lambda[2 \dim X]$, so then $Rf^! \Lambda = Rf^!\omega_X[-2 \dim X] = \omega_Z[-2 \dim X]$. Thus we get a natural map $\alpha: f^\ast \mathcal{F} \otimes \omega_Z[-2 \dim X] \to Rf^! \mathcal{F}$.

Next, note that the complex $\omega_Z$ is concentrated in degrees $[-2 \dim Z,0]$, and in the lowest of these degrees it’s just the constant sheaf, i.e. $\tau^{ \leq -2 \dim Z} \omega_Z \cong \Lambda[2 \dim Z]$. In particular, there is a canonical map $\Lambda[2 \dim Z] \to \omega_Z$. Shifting by $-2 \dim X$ and tensoring with $f^\ast \mathcal{F}$ gives a map $\beta: f^\ast \mathcal{F}[2 \dim Z - 2 \dim X] \to f^\ast \mathcal{F} \otimes \omega_Z[-2 \dim X]$. Putting things together, we get a natural map $\alpha \circ \beta: f^\ast \mathcal{F}[2 \dim Z - 2 \dim X] \to Rf^! \mathcal{F}$. Set $c= \dim X - \dim Z,$ so after shifting this becomes a natural map

$\gamma=\gamma_{\mathcal{F}}: f^{\ast}\mathcal{F}[-c] \to Rf^!\mathcal{F}[c]$.

This shifting has the advantage that Verdier duality exchanges the functors $f^{\ast}[-c]$ and $Rf^![c]$ on $D^b_c$, and one can check that the Verdier dual of $\gamma_{\mathcal{F}}$ identifies with $\gamma_{\mathbf{D}\mathcal{F}}$.

Definition. The functor $f^{\ast !}: \mathrm{Perv}(X) \to \mathrm{Perv}(Z)$ sends any $\mathcal{F}$ to the image of the map $^p\mathcal{H}^0(\gamma):$ $^p\mathcal{H}^0(f^{\ast} \mathcal{F}[-c]) \to$ $^p\mathcal{H}^0(Rf^!\mathcal{F}[c])$.

Here of course $^p\mathcal{H}^0(-)$ denotes the zeroth perverse cohomology sheaf.

Exercise. Show that $f^{\ast !}(\Lambda[\dim X]) \cong \mathcal{IC}_Z$.

It might be interesting to compute this functor in some other examples. Note that it can be quite stupid: if $f$ is a closed immersion (with $c > 0$) and $\mathcal{F} \in \mathrm{Perv}(X)$ is already supported on $Z$, then $f^{ \ast !} \mathcal{F} = 0$. On the other hand, if $f$ is smooth and surjective, then $f^{\ast !} \cong f^{\ast}[-c] \cong Rf^![c]$ is faithful.

## Report from Oberwolfach

Just returned from a workshop on “Arithmetic of Shimura varieties” at Oberwolfach. Some scattered recollections:

• Gabber wasn’t there, but there were some Gabberesque moments anyway. In particular, during Xuhua He’s talk, Goertz observed that a point is an example of a Deligne-Lusztig variety, so any variety is a union of Deligne-Lusztig varieties. Gotta be careful…
• The food was about the same as usual. Worst Prize was tied between two dishes: a depressing vegetable soup which somehow managed to be flavorless and bitter simultaneously, and a dessert which looked like a lovely innocent custard but tasted like balsamic vinegar. The best dishes were all traditional German fare.
• Best Talks (in no particular order): Jean-Stefan Koskivirta, Miaofen Chen, Ben Howard, Timo Richarz.
• Apparently this paper can be boiled down to a page or two.
• There was (not surprisingly) some late-night discussion of the Stanford Mystery. [Redacted] proposed a theory so mind-bogglingly outrageous that it certainly won’t fit in this margin.
• “Fun was never really my goal.” – A representative UChicago alum.
• On Thursday it snowed, and a snowball fight broke out after dinner. This was a lot of fun, but I’m still glad we didn’t follow Pilloni’s suggestion of a match between Team Europe (Pilloni, Stroh, Morel, Anschutz, Richarz, Mihatsch, etc.) and Team USA (me).
• Here’s an innocent problem which turns out to be pretty tricky. Let $X$ be a (separated, smooth) rigid analytic space over $\mathbf{Q}_p$, and let $Y \to X$ be a map from a perfectoid space which is a $\underline{G}$-torsor for some profinite group $G$. In shorthand, you should think that $X =" Y/G$ with G acting freely (this is all literally true in the category of diamonds). It’s easy to cook up examples of this scenario: for instance, you can take $X=\mathrm{Spa}\mathbf{C}_p \left\langle T^{\pm 1} \right\rangle$ and $Y=\mathrm{Spa}\mathbf{C}_p \left\langle T^{\pm 1/p^\infty} \right\rangle$, so then $Y \to X$ is a torsor for the group $\mathbf{Z}_p$. However, there are also much more complicated examples which arise in nature. In particular, if $X$ is a Rapoport-Zink space or abelian-type Shimura variety at some finite level, and $Y$ is the associated infinite level perfectoid guy over it, then we’re in the situation above, with $G$ open in the $\mathbf{Q}_p$-points of some auxiliary reductive group.

Anyway, supposing we’re in the situation above, we can ask the following complementary questions:
Q1. Suppose that $Y$ is affinoid perfectoid. Does this imply that $X$ is an affinoid rigid space?
Q2. Suppose that $X$ is an affinoid rigid space. Does this imply that $Y$ is affinoid perfectoid?

It seems like both of these questions are actually really hard! For Q1, we can (by assumption) write $Y=\mathrm{Spa}(A,A^+)$ for some perfectoid Tate-Huber pair $(A,A^+)$, and then one might guess that $X$ coincides with $X'=\mathrm{Spa}(A^G,A^{+G})$. There is certainly a map $X \to X'$, but now one is faced with the problem of showing that $A^G$ is “big enough” for this map to be an isomorphism. This can be reduced to any one of a handful of auxiliary problems, but they all seem hard (at least to me). For instance, as a warmup one could try to prove either of the following implications:

W1. Under the hypothesis of Q1, $H^1(X,\mathcal{O}_X)$ vanishes.
W2. Under the hypothesis of Q1, $H^1(X,\mathcal{O}_{X}^+)$ is killed by a fixed power of $p.$

Both of these conclusions would certainly hold if we already knew that $X$ was affinoid: the first is just (a consequence of) Tate acyclicity, while the fact that $H^1(X,\mathcal{O}_{X}^+)$ is killed by some power of $p$ for smooth affinoids is a non-trivial theorem of Bartenwerfer. But I have totally failed to prove either W1 or W2.

In any case, the essential point with Q1 seems to be the following. If $H$ is some open subgroup, then $(A^+/p^n)^H$ will always have plenty of elements, and indeed taking the direct limit as $H$ shrinks recovers $A^+/p^n$. But the obstruction to lifting an element of $(A^+/p^n)^H$ to an element of $(A^+)^H$ is a torsion class in $H^1(H,A^+)$, and the latter group seems hard to control.

For Q2, there is maybe a slightly clearer path through the forest: it would follow from the following conjecture, which I explained during my talk in the workshop.

To set things up, let $(A,A^+)$ be any uniform Tate-Huber pair over $(\mathbf{Q}_p,\mathbf{Z}_p)$, and let $X=\mathrm{Spa}(A,A^+)$ be the associated pre-adic space. Let $X_v$ denote the site given by perfectoid spaces over $X$ with covers given by v-covers, and let $\mathcal{O}$ and $\mathcal{O}^+$ be the obvious structure sheaves on $X_v$. Set $\breve{A}^+ = H^0(X_v,\mathcal{O}^+)$ and $\breve{A} = \breve{A}^+ [1/p] = H^0(X_v,\mathcal{O})$, so the association $(A,A^+) \mapsto (\breve{A}, \breve{A}^+)$ is an endofunctor on the category of uniform Tate-Huber pairs over $\mathbf{Q}_p$. One can check that breve’ing twice is the same as breve’ing once, and that the natural map $(A,A^+) \to (\breve{A},\breve{A}^+)$ induces an isomorphism of diamonds. If $A$ is a smooth (or just seminormal) affinoid $K$-algebra for some $K/\mathbf{Q}_p$, or if $A$ is perfectoid, then breve’ing doesn’t change $A$.

Conjecture. Let $(A,A^+)$ be a uniform Tate-Huber pair over $\mathbf{Q}_p$ such that every completed residue field of $\mathrm{Spa}(A,A^+)$ is a perfectoid field. Then $\breve{A}$ is a perfectoid Tate ring.

Aside from disposing of Q2, this conjecture would settle another notorious problem: it would imply that if $A$ is a uniform sheafy Huber ring and $\mathrm{Spa}(A,A^+)$ is a perfectoid space, then $A$ is actually perfectoid.

It may be instructive to see an example of a non-perfectoid (uniform) Tate ring which satisfies the hypothesis of this conjecture. To make an example (with $p>2$), set $A=\mathbf{C}_p \left\langle T^{1/p^\infty} \right\rangle$, and let $B=A[\sqrt{T}]$ with the obvious topology. Set $C=\mathbf{C}_p \left\langle T^{1/2p^\infty} \right\rangle$, so there are natural maps $A \to B \to C$. Then $A$ and $C$ are perfectoid, but $B$ isn’t: the requisite $p$-power roots of $\sqrt{T}$ mod $p$ don’t exist. Nevertheless, every completed residue field of $B$ is perfectoid (exercise!), and the map $B \to C$ induces an isomorphism $\breve B \cong C$.

OK, this bullet point turned out pretty long, but these things have been in my head for the last couple months and it feels good to let them out. Besides, Yoichi Mieda asked me about Q1 during the workshop, so despite the technical nature of these questions, I might not be the only one who cares.

• Oberwolfach continues to be one of the best places in the world to do mathematics.

Thanks to the organizers for putting together such an excellent week!

## Artin-Grothendieck vanishing, again

A few years ago I started thinking about whether there was a natural rigid analytic version of the Artin-Grothendieck vanishing theorem. Last summer this grew into an obsession, and I managed to prove some general results. In particular, I showed that if $X$ is an affinoid rigid space over a complete algebraically closed field, AND $X$ comes via base change from an affinoid defined over a discretely valued subfield, then $H^i(X,\mathbf{Z}/n)=0$ for all $i > \mathrm{dim}(X)$ and all $n$ prime to the residue characteristic. I also proved a similar result with a non-constant coefficient sheaf, assuming moreover that the base field is of characteristic zero. This all got written up here.

Now, the hypothesis of definability over a discretely valued field is clearly stupid and shouldn’t be there, but I wasn’t able to remove it. So I was extremely happy this morning when Akhil Mathew and Bhargav Bhatt sent me an expanded version of their paper on arc-descent, in which they give a beautiful proof of rigid analytic Artin-Grothendieck vanishing without any superfluous assumptions. Their arguments are phrased in terms of algebraic geometry, rather than rigid analysis; in this post I want to recast (mostly for my own benefit I guess) the essential point of their argument in rigid analytic language.

The key is to prove the following.

Theorem (Bhatt-Mathew). Let $\mathrm{Spa}A$ be an affinoid rigid space over a complete algebraically closed nonarchimedean field $K$. Set $\Lambda = \mathbf{Z}/n$ where $n$ is any integer prime to the residue characteristic. Then $R\Gamma(\mathrm{Spa}A,\Lambda) \in D^{\leq \mathrm{dim}A}(\Lambda)$.

This implies the characteristic zero case of Conjecture 1.2 in my paper.

The proof proceeds in three steps.

Step One: Treat the case where $\mathrm{Spa}A$ is smooth. This was already done by Berkovich in the 90’s and I’ll take it for granted, although BM give their own nice argument for it. (Both arguments eventually appeal to the classical Artin-Grothendieck vanishing theorem.)

Step Two: Prove the weaker statement that $R\Gamma(\mathrm{Spa}A,\Lambda) \in D^{\leq 1+\mathrm{dim}A}(\Lambda)$ in general.

For this we use induction on $\mathrm{dim}A$. I’ll assume for simplicity that $K$ has characteristic zero. Without loss of generality we can assume that $A$ is reduced. Then by excellence of affinoid algebras, we can pick some non-zero-divisor $f \in A$ such that $A[1/f]$ is regular. Fix a nonzero nonunit $\pi \in \mathcal{O}_K$, and for any $n \geq 1$ consider the rational subsets $U_n = \{ x\,with\,|f(x)| \geq |\pi|^n \}$ and $V_n = \{x\,with\,|f(x)| \leq |\pi|^n \}$ inside $\mathrm{Spa}A$. Set $W_n = U_n \cap V_n$, so we get a Mayer-Vietoras distinguished triangle

$R\Gamma(\mathrm{Spa}A,\Lambda) \to R\Gamma(U_n,\Lambda)\oplus R\Gamma(V_n,\Lambda) \to R\Gamma(W_n,\Lambda)\to$

for any $n$. Note that $U_n$ and $W_n$ are smooth affinoids, so their etale cohomology is concentrated in degrees $\leq \mathrm{dim}A$ by Step One. Therefore, truncating the above Mayer-Vietoras sequence we get a quasi-isomorphism

$\tau^{\geq \mathrm{dim}A+2}R\Gamma(\mathrm{Spa}A,\Lambda) \simeq \tau^{\geq \mathrm{dim}A+2}R\Gamma(V_n,\Lambda)$

for any $n$. Moreover, $\mathrm{Spa}(A/f) \sim \lim_{n} V_n$ in the sense of adic spaces, which implies that the etale cohomology of the left-hand side is the colimit of the etale cohomologies of the right-hand sides. Therefore, passing to the colimit over $n$, the previous quasi-isomorphism gives a quasi-isomorphism

$\tau^{\geq \mathrm{dim}A+2} R\Gamma(\mathrm{Spa}A,\Lambda) \simeq \tau^{\geq \mathrm{dim}A+2}R\Gamma(\mathrm{Spa}(A/f),\Lambda)$.

But now we win, because $A/f$ is an affinoid of dimension $\dim(A)-1$, so by the induction hypothesis its etale cohomology is concentrated in degrees $\leq \mathrm{dim}A$.

Step Three. Bootstrap from the result of Step Two by a trick. More precisely, let $X=\mathrm{Spa}A$ and $\Lambda=\mathbf{Z}/n$ be as in the statement of the main theorem. By Step Two, we just have to show that $H^{\mathrm{dim}+1}(X,\Lambda)=0$. By another application of Step Two, the complex $R\Gamma(X,\Lambda) \otimes_{\Lambda}^{\mathbf{L}} R\Gamma(X,\Lambda)$ has cohomology in degree $2\mathrm{dim}A+2$ given by $H^{\mathrm{dim}A+1}(X,\Lambda)^{\otimes 2}$, and its enough to show that the latter module is zero. But

$R\Gamma(X,\Lambda) \otimes_{\Lambda}^{\mathbf{L}} R\Gamma(X,\Lambda) \simeq R\Gamma(X \times X,\Lambda)$

by the Kunneth formula*, and $X \times X$ is a $2\mathrm{dim}A$-dimensional affinoid, so its cohomology is concentrated in degrees $\leq 2\mathrm{dim}A+1$ by yet another application of Step Two. This gives the result.

*The necessary result is that if $X$ and $Y$ are $K$-affinoid spaces, then $R\Gamma(X \times Y, \Lambda) \simeq R\Gamma(X,\Lambda) \otimes_{\Lambda}^{\mathbf{L}} R\Gamma(Y,\Lambda)$. I’m not sure if this is in the literature; Bhargav and Akhil prove (an algebraic form of) it in their paper. However, it is easy to deduce this from the results in Huber’s book. The point is that $X, Y$ have canonical adic compactifications $\overline{X},\overline{Y}$, and etale cohomology (with constant coefficients) doesn’t change if you replace $X$ or $Y$ by its compactification. But then $\overline{X}$ and $\overline{Y}$ are proper over $\mathrm{Spa}K$ (in the sense of Huber’s book), so it’s easy to show that

$R\Gamma(\overline{X} \times \overline{Y},\Lambda) \simeq R\Gamma(\overline{X},\Lambda) \otimes_{\Lambda}^{\mathbf{L}} R\Gamma(\overline{Y},\Lambda)$

by the usual combination of proper base change and the projection formula.

## The Newton stratification is true

Let $G$ be a connected reductive group over $\mathbf{Q}_p$, and let $\mu$ be a $G$-valued (geometric) conjugacy class of minuscule cocharacters, with reflex field $E$. In their Annals paper, Caraiani and Scholze defined a very interesting stratification of the flag variety $\mathcal{F}\ell_{G,\mu}$ (regarded as an adic space over $E$) into strata $\mathcal{F}\ell_{G,\mu}^{b}$, where $b$ runs over the Kottwitz set $B(G,\mu^{-1})$. Let me roughly recall how this goes: any (geometric) point $x \to \mathcal{F}\ell_{G,\mu}$ determines a canonical modification $\mathcal{E}_x \to \mathcal{E}_{triv}$ of the trivial $G$-bundle on the Fargues-Fontaine curve, meromorphic at $\infty$ and with “mermorphy $\mu$” in the usual sense. On the other hand, Fargues proved that $G$-bundles on the curve are classified up to isomorphism by $B(G)$, and then Caraiani-Scholze and Rapoport proved that $\mu$-meromorphic modifications of the trivial bundle are exactly classified by the subset $B(G,\mu^{-1})$ (CS proved that only these elements can occur; R proved that all of these elements occur). The Newton stratification just records which element of this set parametrizes the bundle $\mathcal{E}_x$.

The individual strata are pretty weird. For example, if $G=GL_n$ and $\mu=(1,0,\dots,0)$, then $\mathcal{F}\ell_{G,\mu} \simeq \mathbf{P}^{n-1}$ and the open stratum is just the usual Drinfeld space $\Omega^{n-1}$, but the other strata are of the form $\Omega^{n-i-1} \times^{P_{n-i,i}(\mathbf{Q}_p)} GL_n(\mathbf{Q}_p)$, where $P_{n-i,i}$ is the evident parabolic in $GL_n$ and the action on $\Omega^{n-i-1}$ is via the natural map $P_{n-i,i}(\mathbf{Q}_p) \twoheadrightarrow GL_{n-i}(\mathbf{Q}_p)$. Qualitatively, this says that they’re unions of profinitely many copies of lower-dimensional Drinfeld spaces. In particular, the non-open strata are not rigid analytic spaces. There are also examples of strata which don’t have any classical rigid analytic points. However, the $\mathcal{F}\ell_{G,\mu}^{b}$‘s are always perfectly well-defined from the topological or diamond point of view.

Anyway, I’m getting to the following thing, which settles a question left open by Caraiani-Scholze.

Theorem. Topologically, the Newton stratification of $\mathcal{F}\ell_{G,\mu}$ is a true stratification: the closure of any stratum is a union of strata.

The idea is as follows. After base-changing from $E$ to the completed maximal unramified extension $E'$ (which is a harmless move), there is a canonical map $\zeta: \mathcal{F}\ell_{G,\mu,E'} \to \mathrm{Bun}_{G}$ sending $x$ to the isomorphism class of $\mathcal{E}_x$. Here $\mathrm{Bun}_{G}$ denotes the stack of $G$-bundles on the Fargues-Fontaine curve, regarded as a stack on the category of perfectoid spaces over $\overline{\mathbf{F}_p}$. This stack is stratified by locally closed substacks $\mathrm{Bun}_{G}^{b}$ defined in the obvious way, and by construction the Newton stratification is just the pullback of this stratification along $\zeta$. Now, by Fargues’s theorem we get an identification $|\mathrm{Bun}_{G}| = B(G)$, so it is completely trivial to see that the stratification of $\mathrm{Bun}_{G}$ is a true stratification (at the level of topological spaces). We then conclude by the following observation:

Proposition. The map $\zeta$ is universally open.

The idea is to observe that $\zeta$ factors as a composition of two maps $\mathcal{F}\ell_{G,\mu,E'} \to [\mathcal{F}\ell_{G,\mu,E'}/\underline{G(\mathbf{Q}_p)}] \to \mathrm{Bun}_{G}$. Here the first map is a $\underline{G(\mathbf{Q}_p)}$-torsor by construction, so it’s universally open by e.g. Lemma 10.13 here. More subtly, the second map is also universally open. Why? Because it is cohomologically smooth in the sense of Definition 23.8 here; universal openness then follows by Proposition 23.11 in the same document.

For the cohomological smoothness claim, take any affinoid perfectoid space with a map $T \to \mathrm{Bun}_{G}$, corresponding to some bundle $\mathcal{F} / \mathcal{X}_T$. After some thought, one works out the fiber product $X = T \times_{\mathrm{Bun}_{G}} [\mathcal{F}\ell_{G,\mu,E'}/\underline{G(\mathbf{Q}_p)}]$ “explicitly”: it parametrizes untilts of $T$ over $E'$ together with isomorphism classes of $\mu^{-1}$-meromorphic modifications $\mathcal{E}\to \mathcal{F}$ supported along the section $T^{\sharp} \to \mathcal{X}_T$ induced by our preferred untilt, with the property that $\mathcal{E}$ is trivial at every geometric point of $T$. Without the final condition, we get a larger functor $X'$ which etale-locally on $T$ is isomorphic to $T \times_{\mathrm{Spd}(\overline{\mathbf{F}_p})} \mathcal{F}\ell_{G,\mu^{-1},E'}^{\lozenge}$. (To get the latter description, note that etale-locally on $T$ we can trivialize $\mathcal{F}$ on the formal completion of the curve along $T^{\sharp}$, and then use Beauville-Laszlo to interpret the remaining data as a suitably restricted modification of the trivial $G$-torsor on $\mathrm{Spec} \mathbf{B}_{dR}^{+}(\mathcal{O}(T^{\sharp}))$. This is a Schubert cell in a Grassmannian. Then use Caraiani-Scholze’s results on the Bialynicki-Birula map.) Anyway anyway, after a little more fiddling around the point is basically that the projection $X' \to T$ is cohomologically smooth because it’s the base change of a smooth map of rigid spaces. By Kedlaya-Liu plus epsilon, the natural map $X \to X'$ is an open immersion, so $X \to T$ is cohomologically smooth. Since $T$ was arbitrary, this is enough.

## A counterexample

Let $C/\mathbf{Q}_p$ be a complete algebraically closed nonarchimedean field extension, and let $X$ be any proper rigid space over $C$. Let $\mathbf{L}$ be any $\mathbf{Z}_p$-local system on $X_{\mathrm{proet}}$. By the main results in Scholze’s p-adic Hodge theory paper, the pro-etale cohomology groups $H^i_{\mathrm{proet}}(X,\mathbf{L})$ are always finitely generated $\mathbf{Z}_p$-modules. It also seems likely that Poincare duality holds in this setting (and maybe someone has proved this?).

Suppose instead that we’re given a $\mathbf{Q}_p$-local system $\mathbf{V}$. By analogy, one might guess that the cohomology groups $H^i_{\mathrm{proet}}(X,\mathbf{V})$ are always finitely generated $\mathbf{Q}_p$-vector spaces. Indeed, this (and more) was claimed as a theorem by Kedlaya-Liu in a 2016 preprint. However, it is false. The goal of this post is to work out an explicit counterexample.

So, consider $X=\mathbf{P}^1$ as a rigid space over $C$. This is the target of the Gross-Hopkins period map $\pi_{\mathrm{GM}}: \mathcal{M} \to X$, where $\mathcal{M}$ is (the rigid generic fiber of the base change to $\mathcal{O}_C$ of) the Lubin-Tate deformation space of some fixed connected p-divisible group $G_0/\overline{\mathbf{F}_p}$ of dimension 1 and height 2. The rational p-adic Tate module of the universal p-divisible group $G/\mathcal{M}$ descends along $\pi_{\mathrm{GM}}$ to a rank two $\mathbf{Q}_p$ local system $\mathbf{V}_{LT}$ on $X$.

Theorem. Maintain the above setup. Then
i. For any $i \neq 1,2$ the group $H^i_{\mathrm{proet}}(X,\mathbf{V}_{LT})$ is zero.

ii. The group $H^1_{\mathrm{proet}}(X,\mathbf{V}_{LT})$ is a Banach-Colmez space over $C$ of Dimension $(1,-2)$.
iii. The group $H^2_{\mathrm{proet}}(X,\mathbf{V}_{LT})$ is a Banach-Colmez space over $C$ of Dimension $(1,2)$.

Recall that a Banach-Colmez space is a special kind of topological $\mathbf{Q}_p$-vector space (really it’s a functor valued in such things, but I’ll be a little sloppy about this point). Morally, it’s something like a finite-dimensional $C$-vector space defined up to a finite-dimensional $\mathbf{Q}_p$-vector space. In particular, any such space has a well-defined Dimension, which is a pair in $\mathbf{Z}_{\geq 0} \times \mathbf{Z}$ whose entries record the $C$-dimension and the $\mathbf{Q}_p$-dimension of the space, respectively. So for example the space $C^2$ has Dimension $(2,0)$, and the space $C/\mathbf{Q}_p$ has Dimension $(1,-1)$. Unsurprisingly, any Banach-Colmez space whose $C$-dimension is positive will be disgustingly infinitely generated as a $\mathbf{Q}_p$-vector space, so the Theorem really does give us an example of the desired type. Note also that Poincare duality fails in this example.

Proof. Let $\mathbf{B}_{\mathrm{crys}}^{+,\varphi^2=p}$ be the evident sheaf on $X_{\mathrm{proet}}$, where e.g. $\mathbf{B}_{\mathrm{crys}}^{+}$ is the crystalline period sheaf defined in Tan-Tong’s paper on crystalline comparison. The key observation is that there is a short exact sequence

$(\ast)\;\;\; 0 \to \mathbf{V}_{LT} \to \mathbf{B}_{\mathrm{crys}}^{+,\varphi^2=p} \to \mathrm{Lie}(G)[\tfrac{1}{p}] \otimes_{\mathcal{O}_X} \widehat{\mathcal{O}}_X \to 0$

of sheaves on $X_{\mathrm{proet}}$. This is a sheaf-theoretic version of the sequence (1.0.1) in Scholze-Weinstein, and it can be constructed using the methods in their paper. One can also construct it directly using the modern fancy-pants interpretation of $X$ in this setup as the period domain parametrizing admissible length one modifications of the bundle $\mathcal{O}(1/2)$ on the Fargues-Fontaine curve. (Nb. The mysterious middle term in the sequence is really the sheaf of $\varphi$-equivariant maps from $H^1_{\mathrm{crys}}(G_0/W(\overline{\mathbf{F}_p}))[\tfrac{1}{p}]$ to $\mathbf{B}_{\mathrm{crys}}^{+}$.)

Anyway, this reduces us to computing the groups $H^i_{\mathrm{proet}}(X,\mathbf{B}_{\mathrm{crys}}^{+,\varphi^2=p})$ and $H^i_{\mathrm{proet}}(X,\mathrm{Lie}(G)[\tfrac{1}{p}] \otimes_{\mathcal{O}_X} \widehat{\mathcal{O}}_X)$. This might look a bit terrifying, but it’s not so bad. In fact, the first of these can be handled by a general lemma.

Lemma. Let $\mathbf{M}$ be any Banach-Colmez space over $C$. For any proper rigid space $X/C$, we may regard $\mathbf{M}$ as a (pre)sheaf on $X_{\mathrm{proet}}$, so in particular we can talk about the pro-etale cohomology groups $H^i_{\mathrm{proet}}(X,\mathbf{M})$. In this notation, the natural map $H^i_{\mathrm{proet}}(X,\mathbf{Q}_p) \otimes_{\mathbf{Q}_p} \mathbf{M}(C) \to H^i_{\mathrm{proet}}(X,\mathbf{M})$ is an isomorphism.

(Proof sketch: Use the 5 lemma to reduce to the case of effective Banach-Colmez spaces, and then to the cases of the space $\mathbf{Q}_p$, where it’s a tautology, and the space Colmez notates $\mathbb{V}^1$, where it follows from the primitive comparison theorem, cf. Theorem 3.17 in Scholze’s CDM survey.)

Applied to our problem, this immediately gives that $H^i_{\mathrm{proet}}(X,\mathbf{B}_{\mathrm{crys}}^{+,\varphi^2=p}) \cong H^i_{\mathrm{proet}}(X,\mathbf{Q}_p) \otimes_{\mathbf{Q}_p} B_{\mathrm{crys}}^{+,\varphi^2 = p}$. By the standard easy computation of $H^i_{\mathrm{proet}}(X,\mathbf{Q}_p)$, we get that $H^i_{\mathrm{proet}}(X,\mathbf{B}_{\mathrm{crys}}^{+,\varphi^2=p})$ is a copy of $B_{\mathrm{crys}}^{+,\varphi^2=p}$ for either of $i \in \{ 0,2 \}$, and it vanishes otherwise. In particular, in degrees 0 and 2, it has Dimension $(1,2)$ by some of the calculations in Colmez’s original paper.

Next, we need to compute the pro-etale cohomology of $\mathcal{E} = \mathrm{Lie}(G)[\tfrac{1}{p}] \otimes_{\mathcal{O}_X} \widehat{\mathcal{O}}_X$. For this, we use the fact (already in Gross and Hopkins’s original article) that $\mathrm{Lie}(G)[\tfrac{1}{p}] \simeq \mathcal{O}_X(1)$. Let $\lambda : X_{\mathrm{proet}} \to X_{\mathrm{an}}$ be the evident projection of sites. Combining the description of $\mathrm{Lie}(G)[\tfrac{1}{p}]$ with an easy projection formula gives an isomorphism $E\overset{def}{=}R\lambda_{\ast} \mathcal{E} \cong \mathcal{O}_X (1) \otimes_{\mathcal{O}_X} R\lambda_{\ast} \widehat{\mathcal{O}}_X$. Moreover, $R^i \lambda_{\ast} \widehat{\mathcal{O}}_X \simeq \Omega_{X}^i$ identifies with $\mathcal{O}_X$ in degree zero and $\mathcal{O}_X(-2)$ in degree 1, and vanishes otherwise. In particular, the only nonvanishing cohomology sheaves of $E$ are $\mathcal{O}_X(1)$ in degree 0 and $\mathcal{O}_X(-1)$ in degree 1; moreover, the latter has no global cohomology in any degree. Feeding this information into the Leray spectral sequence for $\lambda$, we get that $H^i_{\mathrm{proet}}(X,\mathcal{E}) \simeq H^i(X,E) \simeq H^i(X, \tau^{\leq 0} E) \simeq H^i(X,\mathcal{O}_X(1))$, so this is $C^2$ for $i=0$ and zero otherwise.

Finally, we can put everything together and take the long exact sequence in pro-etale cohomology associated with $(\ast)$. It’s easy to check that $\mathbf{V}_{LT}$ doesn’t have any global sections, and the middle term of $(\ast)$ has no cohomology in degree one, so we get a short exact sequence $0 \to H^0_{\mathrm{proet}}(X,\mathbf{B}_{\mathrm{crys}}^{+,\varphi^2=p}) \to C^2 \to H^1_{\mathrm{proet}}(X,\mathbf{V}_{LT}) \to 0$. We’ve already identified the $H^0$ here as something of Dimension $(1,2)$, so by the additivity of Dimensions in short exact sequences, we deduce that $H^1_{\mathrm{proet}}(X,\mathbf{V}_{LT})$ has Dimension $(1,-2)$, as desired. By a similar argument, we get an isomorphism $H^2_{\mathrm{proet}}(X,\mathbf{V}_{LT}) \simeq B_{\mathrm{crys}}^{+,\varphi^2=p}$, which we already observed has Dimension $(1,2)$. The vanishing of all the other cohomologies of $\mathbf{V}_{LT}$ also follows easily. $\square$

BTW, there is nothing special about height $2$ in this story; I just stuck with it for convenience. For any heght $h \geq 2$, there is an analogous rank $h$ $\mathbf{Q}_p$-local system $V_{LT,h}$ on $\mathbf{P}^{h-1}$, and one can check that e.g. $H^1_{\mathrm{proet}}(\mathbf{P}^{h-1},\mathbf{V}_{LT,h})$ has Dimension $(h-1,-h)$.

Let me briefly explain the genesis of these calculations. Several months ago Shizhang Li pointed out to me that the primitive comparison theorem doesn’t obviously generalize to $\mathbf{Q}_p$-local systems without globally defined lattices, and he also suggested that the cohomologies of $\mathbf{V}_{LT}$ might be strange. I promptly forgot about this conversation until Monday, when Shizhang reported that Ruochuan Liu had told him in Oberwolfach several weeks ago that the Kedlaya-Liu “theorem” was false. It was then natural to double down on $\mathbf{V}_{LT}$ as a possible counterexample. Immediately after I gave a lecture on the Tan-Tong paper the next day, the sequence $(\ast)$ entered my head randomly. Then everything just came out. Also, luckily, we were at a restaurant with paper tablecloths.

(I’m not sure whether Ruochuan also had this particular counterexample in mind.)

There are lots of interesting questions here, I think. Are there other natural examples of this type? Are the pro-etale cohomology groups of $\mathbf{Q}_p$-local systems on proper rigid spaces always Banach-Colmez spaces?

## Extending finite etale coverings

Fix a nonarchimedean field $K$ of residue characteristic $p$, and let $X$ be a normal rigid analytic space over $K$.  Suppose we’re given a closed nowhere-dense analytic subspace $Z \subset X$ and a finite etale cover $Y$ of $X \smallsetminus Z$.  It’s natural to ask if $Y$ can be extended to a finite cover $Y' \to X$, and whether some further conditions on $Y'$ pin such an extension down uniquely.  Although the analogous problem for complex analytic spaces was solved by Stein and Grauert-Remmert in the 50s (cf. Grauert and Remmert’s article here), there isn’t very much literature on this problem in the rigid analytic context, with the notable exception of Lutkebohmert’s paper, about which more in a minute.  Anyway, it turns out that at least for a base field $K$ of characteristic zero, this problem has a very satisfying answer, and the proof is a fun exercise in swinging lots of big hammers.

First, here’s the precise definition of “cover” which we’ll use.

Definition. Let $X$ be a normal rigid analytic space.  A cover of $X$ is a finite surjective map $\pi: Y \to X$ from a normal rigid space $Y$, such that one of the following two equivalent conditions holds:
1. There exists a closed nowhere-dense analytic subset $Z \subset X$ such that $\pi^{-1}(Z)$ is nowhere-dense and $Y \smallsetminus \pi^{-1}(Z) \to X \smallsetminus Z$ is finite etale.
2.  Each irreducible component $Y_i$ of $Y$ maps surjectively onto an irreducible component $X_i$ of $X$, and contains a point $y_i$ such that $\mathcal{O}_{X,\pi(y_i)} \to \mathcal{O}_{Y_i,y_i}$ is etale.

Equivalence of these conditions is a fun exercise left to the reader; note that the second requirement in 2. is automatic when $K$ has characteristic 0.

Theorem. Let $X$ be a normal rigid analytic space over a characteristic zero nonarchimedean field $K$, and let $Z \subset X$ be any closed nowhere-dense analytic subset.  Then any finite etale cover of $X \smallsetminus Z$ extends uniquely to a cover of $X$.

In other words, the restriction functor from {covers of $X$ etale over $X \smallsetminus Z$} to {finite etale covers of $X \smallsetminus Z$} is an equivalence of categories.

The uniqueness holds without any condition on $K$, and is an easy consequence of a powerful theorem due to Bartenwerfer.  To explain this result, let $X$ be a normal rigid space and let $Z \subset X$ be any closed nowhere-dense analytic subset. Then Barternwerfer proved that any bounded function on $X \smallsetminus Z$ extends (uniquely) to a function on $X$. In particular, if $Y \to X$ is a cover and $U \subset X$ is any open affinoid subset, then $\mathcal{O}_Y(\pi^{-1}(U)) \cong \mathcal{O}_{Y}^{+}(\pi^{-1}(U \smallsetminus U \cap Z))[1/ \varpi]$ depends only on the restriction of $Y$ to $X \smallsetminus Z$. Since the affinoids $\pi^{-1}(U)$ cover $Y$, this gives the desired uniqueness.  More generally, this argument shows that for any closed nowhere-dense analytic subset $Z \subset X$, the restriction functor from covers of $X$ to covers of $X \smallsetminus Z$ is fully faithful.

The existence of an extension is harder, of course.  Until further notice, assume $K$ has characteristic zero.  Note that by the uniqueness argument, we can always work locally on $X$ when extending a finite etale cover of $X \smallsetminus Z$.  Now the key input is the following base case, due to Lutkebohmert:

Theorem (Lutkebohmert): If $X$ is a smooth rigid space and $Z \subset X$ is a simple normal crossings divisor, then any finite etale cover of $X \smallsetminus Z$ extends to a cover of $X$.

This is more or less an immediate consequence of Lemma 3.3 in Lutkebohmert’s paper, although he doesn’t state this result so explicitly (and curiously, he never discusses the uniqueness of extensions).  The main ingredient aside from this Lemma is a result of Kiehl on “tubular neighborhoods”, which says (among other things) that if $D \subset X$ is a snc divisor in a smooth rigid space, then for any point $x$ in $D$ at which $r$ components of $D$ meet, we can find some small affinoid neighborhood $U$ of $x$ in $X$ together with a smooth affinoid $S$ and an isomorphism $U \simeq S \times B^r$ (where $B^r = \mathrm{Sp}K \left\langle X_1, \dots, X_r \right\rangle$ denotes the r-dimensional closed ball) under which the individual components of $D$ meeting $x$ identify with the zero loci of the coordinate functions $X_i$.

Granted these results, we argue as follows.  Clearly we can assume that $X$ is quasicompact.  We now argue by induction on the maximal number $i(D)$ of irreducible components of $D$ passing through any individual point of $X$. Let me sketch the induction informally. If $i(D)=1$, then $D$ is smooth, so Kiehl’s result puts us exactly in the situation covered by the case $r=1$ of Lemma 3.3. If $i(D)=2$, then locally on $X$ we can assume that $D$ has two smooth components $D_1$ and $D_2$. By the previous case, any finite etale cover $Y$ of $X \smallsetminus D$ extends uniquely to covers $Y_i$ of $X \smallsetminus D_i$, which then glue to a cover $Y_0$ of $X \smallsetminus D_1 \cap D_2$.  But now locally along $D_1 \cap D_2$, Kiehl’s result puts is in the situation covered by the case $r=2$ of Lemma 3.3, and then $Y_0$ extends to a cover of $X$.  If $i(D)=3$, then locally on $X$ we can assume that $D$ has three smooth components $D_1, D_2, D_3$. By the previous case, any finite etale cover of $X \smallsetminus D$ extends to a cover $Y_i$ of $X \smallsetminus D_i$, for each $i \in \{1,2,3\}$; here we use the fact that $i(D \smallsetminus D_i) \leq 2$ for $D \smallsetminus D_i$ viewed as a strict normal crossings divisor in $X \smallsetminus D_i$.  Again the $Y_i$‘s glue to a cover $Y_0$ of $X \smallsetminus D_1 \cap D_2 \cap D_3$, and again locally along $D_1 \cap D_2 \cap D_3$ Kiehl’s result puts us in the situation handled by Lemma 3.3, so $Y_0$ extends to a cover of $X$.  Etc.

To get existence in the general case, we use some recent results of Temkin on resolution of singularities.  More precisely, suppose $X = \mathrm{Sp}(A)$ is an affinoid rigid space, and $Z \subset X$ is a closed nowhere-dense subset as before; note that $Z=\mathrm{Sp}(B)$ is also affinoid, so we get a corresponding closed immersion of schemes $\mathcal{Z} = \mathrm{Spec}(B) \to \mathcal{X} = \mathrm{Spec}(A)$.  These are quasi-excellent schemes over $\mathbf{Q}$, so according to Theorem 1.1.11 in Temkin’s paper, we can find a projective birational morphism $f: \mathcal{X}' \to \mathcal{X}$ such that $\mathcal{X}'$ is regular and $(\mathcal{X}' \times_{\mathcal{X}} \mathcal{Z})^{\mathrm{red}}$ is a strict normal crossings divisor, and such that $f$ is an isomorphism away from $\mathcal{Z} \cup \mathcal{X}^{\mathrm{sing}}$.  Analytifying, we get a proper morphism of rigid spaces $g: X' \to X$ with $X'$ smooth such that $g^{-1}(Z)^{\mathrm{red}}$ is an snc divisor etc.

Suppose now that we’re given a finite etale cover $Y$ of $X \smallsetminus Z$.   Pulling back along $g$, we get a finite etale cover of $X' \smallsetminus g^{-1}(Z)$, which then extends to a cover $h: Y'\to X'$ by our previous arguments. Now, since $g \circ h$ is proper, the sheaf $(g \circ h)_{\ast} \mathcal{O}_{Y'}$ defines a sheaf of coherent $\mathcal{O}_X$-algebras. Taking the normalization of the affinoid space associated with the global sections of this sheaf, we get a normal affinoid $Y''$ together with a finite map $Y'' \to X$ and a canonical isomorphism $Y''|_{(X \smallsetminus Z)^{\mathrm{sm}}} \cong Y|_{(X \smallsetminus Z)^{\mathrm{sm}}}$. The cover we seek can then be defined, finally, as the Zariski closure $Y'''$ of $Y''|_{(X \smallsetminus Z)^{\mathrm{sm}}}$ in $Y''$: this is just a union of irreducible components of $Y''$, so it’s still normal, and it’s easy to check that $Y'''$ satisfies condition 1. in the definition of a cover. Finally, since $Y'''$ and $Y$ are canonically isomorphic after restriction to $(X \smallsetminus Z)^{\mathrm{sm}}$, the uniqueness argument shows that this isomorphism extends to an isomorphism $Y'''|_{X \smallsetminus Z} \cong Y$. This concludes the proof.

Combining this existence theorem with classical Zariski-Nagata purity, one gets a purity theorem for rigid spaces:

Corollary. Let X be a smooth rigid analytic space over a characteristic zero nonarchimedean field, and let $Z \subset X$ be any closed analytic subset which is everywhere of codimension $\geq 2$.  Then finite etale covers of $X$ are equivalent to finite etale covers of $X \smallsetminus Z$.

Presumably this result has other fun corollaries.  I’d be happy to know more.