## 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 $D^{b}_{c}(Y,\mathbf{F}_{\ell})$.

## The one-point compactification of a scheme, part 1

In this series of posts I want to talk about some joint work in progress with Johan de Jong. The goal here is to give a new and totally canonical definition of the functor $Rf_!$ in the etale cohomology of schemes, and to do some really fun geometry along the way. In fact we go beyond the usual situation where $f$ is assumed to be compactifiable, and give a canonical definition of $Rf_!$ for $f$ any locally finite type morphism of locally Noetherian schemes. Such a definition has also been proposed by Liu-Zheng, by a method which requires a very heavy use of $\infty$-categories. Our approach does not involve $\infty$-categories at all.

For motivation, let $X$ be a locally compact Hausdorff space. The one-point compactification of $X$ is obtained by suitably topologizing the set $\overline{X}=X \cup \{\infty\}$: precisely, one takes the open subsets to be all the open subsets of $X$ together with all subsets of the form $V \cup \{\infty \}$ where $V \subset X$ is such that $X \smallsetminus V$ is a closed compact subset of $X$. Then $\overline{X}$ is a compact Hausdorff space, and $X$ is a dense open subset of $\overline{X}$ if $X$ is non-compact. Quite generally, one can check that the inclusion $X \to \overline{X}$ is final among all open embeddings of locally compact Hausdorff spaces $X \to Y$.

Our first goal is to imitate this compactification in the setting of schemes. The correct definition turns out to be as follows.

Definition 1.1. Fix a base scheme $S$, and let $f: X \to S$ be a morphism of schemes which is separated and of finite type (for brevity, we say $X$ is a good $S$-scheme if its structure map is separated and of finite type). The one-point compactification of $X$ over $S$, denoted $\overline{X}^S$, is the contravariant functor from $S$-schemes to sets sending an $S$-scheme $T\to S$ to the set of closed subschemes $Z \subset X_T = X\times_S T$ such that the composite map $Z \to X_T \to T$ is an open immersion. Equivalently, $\overline{X}^S(T)$ is the set of pairs $(Z,\varphi)$ where $Z \subset T$ is an open subscheme and $\varphi : Z \to X$ is an $S$-scheme map whose graph $\Gamma_{\varphi}: Z \to X\times_S T$ is a closed immersion.

Usually $S$ will be clear from context, and we’ll abbreviate $\overline{X}^S$ to $\overline{X}$. Let $\overline{f} : \overline{X} \to S$ denote the “structure map”.

(Here and in what follows, we write $\mathrm{Sch}/S$ for the category of $S$-schemes, and we freely “do geometry” in the category of presheaves of sets on $\mathrm{Sch}/S$ in the modern style, since $S$-schemes embed fully faithfully into this category in the usual way. There are some small issues here with “big categories” which I will totally ignore; suffice it to say that we “fix a choice of a category $\mathrm{Sch}/S$” in the sense of the Stacks Project.)

Anyway, here are some immediate observations on this thing. First of all, there is a canonical map $j^X : X \to \overline{X}$ sending a $T$-point $\varphi: T \to X$ to the pair $(T,\varphi)$; indeed, the separateness of $f$ guarantees that $\Gamma_{\varphi} : T \to X_T$ is a closed immersion. Moreover, the structure map $\overline{X} \to S$ has a canonical “section at infinity” $\infty: S \to \overline{X}$ sending any $S$-scheme $T$ to the closed subscheme $Z= \emptyset \subset X_T$, and $j^X$ and $\infty$ are “disjoint” in the evident sense.

Example 1.2. If $S$ is arbitrary and $X \to S$ is proper, then $\overline{X} = X \coprod S$. (Hint: For any $T$-point of $\overline{X}$, the map $Z \to T$ is a proper open immersion.)

Example 1.3. If $S$ is arbitrary and $X= \mathbf{A}^1_S$, then $\overline{X}$ is the ind-scheme obtained as an “infinite pinching” of $\mathbf{P}^1_S$ along the section at infinity. More precisely, we can assume (e.g. by Theorem 1.5.i below) that $S=\mathrm{Spec}A$ is affine. Let $B_n \subset A[t^-1]$ be the ring of polynomials $\sum a_i t^{-i}$ such that $a_i=0$ for all $0 < i < n$. Set  $U=\mathrm{Spec}A[t]$ and $V_n = \mathrm{Spec}B_n$; gluing these along their common open $\mathbf{G}_{m,S}$ in the obvious way, we get an inductive system of schemes $X_1=\mathbf{P}^1_S \to X_2 \to X_3 \to \cdots$, with compatible maps $X_i \to \overline{X}$. In the colimit this gives a map $\mathrm{colim} X_n \to \overline{X}$. This map turns out to be an isomorphism, but this is not so obvious.

Example 1.4. If $S$ is arbitrary and $X = \mathbf{A}^2_S$, then $\overline{X}$ is NOT an ind-scheme or ind-algebraic space.

This last example is typical: for almost all $X \to S$, the functor $\overline{X}$ will not be an object of classical algebraic geometry. Nevertheless, it turns out to have very reasonable properties, which we collect in the following theorem:

Theorem 1.5. Fix a base scheme $S$, and let $X \to S$ and $\overline{X}=\overline{X}^S \to S$ be as above. Then:
i. If $S' \to S$ is any scheme map, there is a canonical isomorphism $\overline{X}^S \times_S S' \cong \overline{X \times_S S'}^{S'}$.
ii. The map $j^X : X \to \overline{X}$ is (representable in schemes and) an open immersion.
iii. The functor $\overline{X}$ is a sheaf for the fpqc topology.
iv. The structure map $\overline{X} \to S$ satisfies the valuative criterion of properness.
v. If $X \to S$ is of finite presentation, then $\overline{X} \to S$ is limit-preserving.
vi. The diagonal $\Delta: \overline{X} \to \overline{X} \times_S \overline{X}$ is representable in formal algebraic spaces: for any scheme with a map $T \to \overline{X} \times_S \overline{X}$, the fiber product $W= \overline{X} \times_{\Delta, \overline{X} \times_S \overline{X}} T$ is representable by a formal algebraic space whose underlying reduced subspace is a closed subscheme of $T$. If $S$ is locally Noetherian then $W$ is a countably indexed directed colimit of closed subschemes of $T$ along thickenings.
vii. If $g: X \to Y$ is any proper map of good $S$-schemes, there is a canonical map $\overline{g}: \overline{X} \to \overline{Y}$ such that $j^Y \circ g = \overline{g} \circ j^X$.
viii. If $h: U \to V$ is any open immersion of good $S$-schemes, there is a canonical map $\tilde{h}: \overline{V} \to \overline{U}$ such that $j^U = \tilde{h} \circ j^V \circ h$.

Let me say a few words about the proofs. Part i. is trivial from the definitions. For ii., one checks directly that for a given $T$-point $T \to \overline{X}$ with associated pair $(Z,\varphi)$, the pullback $X \times_{\overline{X}} T$ is just the open subscheme $Z \subset T$.

For iii., let $T' \to T$ be an fpqc cover, and suppose given $(Z',\varphi') \in \overline{X}(T')$ lying in the equalizer of $\overline{X}(T') \rightrightarrows \overline{X}(T' \times_T T')$. One first descends the open subscheme $Z' \subset T'$ to an open subscheme $Z \subset T$ using the fact that $|T'| \to |T|$ is a quotient map, and then one descends the morphism $\varphi'$ to a map $Z \to X$. To see that $(Z,\varphi)$ has the right properties, note that the graph $\Gamma_{\varphi}$ pulls back to the closed immersion $\Gamma_{\varphi'}$ along the fpqc cover $X_{T'} \to X_{T}$, so $\Gamma_{\varphi}$ is necessarily a closed immersion.

For iv., one reduces by i. to checking that if $S=\mathrm{Spec}A$ is the spectrum of an arbitrary valuation ring with generic point $\eta \in S$ and $X \to S$ is any good $S$-scheme, then the evident “restriction” map $r: \overline{X}^S(S) \to \overline{X_{\eta}}^{\eta}(\eta)$ is a bijection. After showing that the points at infinity match up, this reduces to showing that any section $s: \eta \to X_{\eta}$ spreads out to a unique point $(Z,\varphi) \in \overline{X}^S(S)$. For this, let $Z \subset X$ be the scheme-theoretic image of $s$ in $X$. By the Lemma in my previous post, the composite map $Z \to S$ is an open immersion, and we’re done.

For v., one takes an arbitrary $T$-point of $\overline{X}$, where $T = \lim T_i$ is a limit of affine $S$-schemes, and then uses “standard limit nonsense” to show that everything comes by pullback from a $T_i$-point for some $i$ – I leave it as an exercise for the reader to fill in the details, making use of Proposition 8.6.3 and Corollaire 8.6.4 in EGA IV_3.

Part vi. is probably the hardest. Let $T \to \overline{X} \times_{S} \overline{X}$ be as in the statement. This corresponds to a pair of $T$-points of $\overline{X}$, i.e. a pair of closed subschemes $Z_i \subset X_T$ for $i=1,2$ such that the induced maps $Z_i \to T$ are open immersions. Let $U=Z_1 \cup_T Z_2$, so this is an open subscheme of $T$. Let $Z=Z_1 \times_{X_T} Z_2$ be the intersection of the $Z_i$‘s inside $X_T$, so we get natural closed immersions $Z \to Z_i$, and composing either of them with the inclusion $Z_i \to U$ realizes $Z$ as a closed subscheme of the open subscheme $U \subset T$. At this point we make the

Definition. Let $T$ be a scheme, and suppose given an open subscheme $U \subset T$ together with a closed subscheme $Z \subset U$. Let $T_{Z \to U}$ be the subfunctor of $T$ whose $V$-points are given by scheme maps $f: V \to T$ such that $f^{-1}(U) \to U$ factors over the closed immersion $Z \to U$.

Now, returning to the situation at hand, you can check directly from the definitions (if you dare) that the fiber product $W$ in vi. is given by the functor $T_{Z \to U}$, for the specific $T,Z,U$ above. This reduces us to a general result:

Lemma. Notation as in the previous definition, the functor $T_{Z\to U}$ is a formal algebraic space whose underlying reduced subspace is a closed subscheme of $T$, namely the reduced closed subscheme corresponding to the closed subset $|Z| \cup (|T| \smallsetminus |U|) \subset |T|$. If $T$ is Noetherian, $\mathcal{I} \subset \mathcal{O}_T$ is the coherent ideal sheaf corresponding to the scheme-theoretic closure $\overline{Z} \subset T$ of $Z$, and $\mathcal{J}$ is any coherent ideal sheaf whose vanishing locus cuts out the closed subspace $|T| \smallsetminus |U|$, then $T_{Z \to U} \cong \mathrm{colim}\, \underline{\mathrm{Spec}}\mathcal{O_X}/(\mathcal{I}\cdot \mathcal{J}^n)$.

Intuitively, $T_{Z \to U}$ is the “union” inside $T$ of the locally closed subscheme $Z$ and the formal completion of $T$ along the complement of $U$.

For vii., one takes the scheme-theoretic image of $Z \subset X_T$ along the map $Z \to X_T \to Y_T$ and then checks that the resulting closed subscheme $Z' \subset Y_T$ has the right properties; in fact $Z' \simeq Z$.

For viii., one takes the pullback of $Z \subset V_T$ along the open immersion $U_T \to V_T$. This clearly has the right properties.

Note that the contravariant functoriality in part viii. might seem strange at first, since it has no analogue for schematic compactifications.  However, this is totally analogous with the situation for one-point compactifications of topological spaces: if $U \to V$ is an open embedding of locally compact Hausdorff spaces, then $\overline{U}$ is obtained from $\overline{V}$ by contracting $\overline{V} \smallsetminus U$ down to the point at infinity, giving a canonical map $\overline{V} \to \overline{U}$.

In part 2, we’ll discuss the applications to etale cohomology.

## Some disconnected thoughts

Sorry for the silence here, the last six months have been pretty busy. Anyway, here are some small and miscellaneous thoughts.

$\bullet$ Dan Abramovich is a really excellent and entertaining writer.  I especially recommend this BAMS review of books by Cutkosky and Kollar, and this survey article for the 2018 ICM; after reading these, you’ll feel like you really understand what “resolution of singularities” looks like as a working field.  His mathscinet review of Fesenko’s IUTT survey is also deeply funny – let us hope that the phrase “resounding partial success” enters common usage.

$\bullet$ Speaking of the 2018 ICM, Scholze’s article is also now available here. I was amused to see a reference to “prismatic cohomology” in the last paragraph – when I visited Bhargav in October, this object had a truly horrible and useless provisional name, so it’s great to see that name replaced by such a perfect and suggestive one.  (This is a new cohomology theory which is closely related to crystalline cohomology, and is also morally related to diamonds – cf. these notes for the actual definition.)

$\bullet$ Recently I had need of the following result:
Lemma. Let $S=\mathrm{Spec}\,A$ be the spectrum of a valuation ring, with generic point $\eta \in S$. Let $X \to S$ be a separated and finite type map of schemes, and let $x: \eta \to X_{\eta}$ be a section over the generic point of $S$, with scheme-theoretic image $Z \subset X$. Then the induced map $Z \to S$ is an open immersion.

Note that if $X \to S$ is proper, then $Z \to S$ is an isomorphism. Anyway, I ended up working out two proofs: a short argument relying on Nagata compactification, and a longer argument which avoided this result but used some other pretty tricky stuff, including Zariski’s main theorem and Raynaud-Gruson’s magic theorem that if $R$ is a domain, then any flat finite type $R$-algebra is finitely presented. If someone knows a truly elementary proof of this lemma, I’d be interested to see it.

Next time I’ll say something about what I needed this for…

## 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.