## 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!

## Two questions and a story

$\bullet$ Let $f$ be some cuspidal Hecke eigenform, with associated Galois representation $\rho_{f}:G_{\mathbf{Q}}\to \mathrm{GL}_2(\overline{\mathbf{Q}_p})$. A notorious conjecture of Greenberg asserts that if $\rho_{f}|G_{\mathbf{Q}_p}$ is abelian (i.e. is a direct sum of characters), then $f$ is a CM form, or equivalently $\rho_f$ is induced from a character. At some point I was talking about this with Barry Mazur, and he suggested a possible generalization:

Let $\rho:G_{\mathbf{Q}} \to \mathrm{GL}_n(\overline{\mathbf{Q}_p})$ be an irreducible geometric Galois representation. Suppose that $\rho|G_{\mathbf{Q}_p}$ is a direct sum of characters. Is $\rho$ induced from a character?

Emerton found a nice argument which proves Greenberg’s conjecture conditionally on some kind of $p$-adic variational Hodge conjecture. Is there any similar evidence for this question in some higher-dimensional cases?

$\bullet$ Do separated etale maps of schemes satisfy effective descent with respect to fpqc covers? This is known if one restricts to quasi-compact separated etale maps. An analogous result is true for perfectoid spaces.

$\bullet$ Here’s a funny story I heard from Glenn Stevens a while back:

At some point in the early ’90s, before he announced his proof of Fermat, Wiles came to Boston and gave a seminar talk at BU. He spoke about what is now known as the Greenberg-Wiles duality formula. However, he didn’t mention his main motivations for this formula. The upshot is that Stevens came away from the talk with the sad feeling that Wiles had lost his touch.

## Modern fictions

On the website for the journal Algebra and Number Theory, one finds the following remarkable statement:

ANT publishes high-quality articles of interest to a broad readership, at a level surpassing all but the top four or five mathematics journals.

I have no problem with ANT, but shoving this self-assessment down people’s throats strikes me as deeply silly. We can probably all agree that the top five journals are (in some order) JAMS, Annals, Inventiones, Publ. IHES, and Acta Mathematica. So ANT is claiming to surpass Compositio, Duke, Ann. Sci. ENS, Cambridge J. Math, etc. Hmm…

Thanks to [redacted] for pointing this out to me.

## 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?

## A stupid remark on cohomological dimensions

Let $Y$ be a finite-dimensional Noetherian scheme, and let $\ell$ be a prime invertible on $Y$. Gabber proved that if $f:X \to Y$ is any finite type morphism, then there is some integer $N$ such that $R^n f_{\ast} F$ vanishes for all $\ell$-torsion etale sheaves $F$ and all $n > N$, cf. Corollary XVIII.1.4 in the Travaux de Gabber volume. In particular, if the $\ell$-cohomological dimension $\mathrm{cd}_{\ell}(Y)$ is finite, then so is $\mathrm{cd}_{\ell}(X)$. It’s natural to ask for a quantitative form of this implication.

Claim. One has the bound $\mathrm{cd}_{\ell}(X) \leq \mathrm{cd}_{\ell}(Y)+2\mathrm{dim}f+2\mathrm{dim}(X)+s(X)$.

Here $\mathrm{dim}f$ is the supremum of the fibral dimensions of $f$, and $s(X) \in \mathbf{Z}_{\geq 0}$ is defined to be one less than the minimal number of separated open subschemes required to cover $X$. In particular, $s(X)=0$ iff $X$ is separated.

Can this be improved?

Anyway, here’s a cute consequence:

Corollary. Let $Y$ be a finite-dimensional Notherian scheme, and let $\ell$ be a prime invertible on $Y$ such that $\mathrm{cd}_{\ell}(Y) < \infty$. Then all affine schemes $X \in Y_{\mathrm{et}}$ have uniformly bounded $\ell$-cohomological dimension. In particular, the hypotheses of section 6.4 of Bhatt-Scholze hold, so the arguments therein show that $D(Y,\mathbf{F}_{\ell})$ is compactly generated and that its compact objects are exactly the objects of $D^{b}_{c}(Y,\mathbf{F}_{\ell})$.