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.

 

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.