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.

scratch

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

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

 

Extending finite etale coverings

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Diamonds for all!

Regular readers of this blog probably know that I’m obsessed with diamonds.  They can thus imagine my happiness when Peter posted an official foundational reference for diamonds a few weeks ago.

I want to use this occasion to make a remark aimed at graduate students etc. who might be wondering whether they should bother learning this stuff: in my opinion, spending time with difficult* manuscripts like the one above usually pays off in the long run. Of course, this only works if you invest a reasonable amount of time, and there’s some initial period where you’re completely befuddled, but after some months the befuddlement metamorphoses into understanding, and then you have a new set of tools in your toolkit! This shouldn’t be so surprising, though; after all, papers like this are difficult precisely because they are so rich in new ideas and tools.

Really, I’ve had this experience many times now – with the paper linked above and its precedent, with the Kedlaya-Liu “Relative p-adic Hodge theory” series, with Kato’s paper on p-adic Hodge theory and zeta functions of modular forms, etc. – and it was the same every time: for some period of months (or years) I would just read the thing for its own sake, but then at some point something in it would congeal with the rest of the swirling fragments in my head and stimulate me to an idea which never would’ve occurred to me otherwise. It’s the most fun thing in the world. Try it yourself.

 

*Here by “difficult” I don’t mean anything negative, but rather some combination of dense/forbidding/technical – something with a learning curve.  Of course, there are plenty of papers which are difficult for bad reasons, e.g. because they’re poorly written.  Don’t read them.

Nowhere-vanishing sections of vector bundles

Let X/k be a proper variety over some field, and let \mathcal{E} be a vector bundle on X.  The functor of global sections of \mathcal{E}, i.e. the functor sending a scheme f:S \to \mathrm{Spec}\,k to the set H^0(S \times_{k} X, (f \times \mathrm{id})^{\ast} \mathcal{E}), is (representable by) a nice affine k-scheme, namely the scheme \mathcal{S}(\mathcal{E}) = \mathrm{Spec}(\mathrm{Sym}_k H^0(X,\mathcal{E})^{\vee}). Let \mathcal{S}(\mathcal{E})^{\times} \subset \mathcal{S}(\mathcal{E}) denote the subfunctor corresponding to nowhere-vanishing sections s \in H^0(S \times_k X, (f \times \mathrm{id})^{\ast} \mathcal{E}). We’d like this subfunctor to be representable by an open subscheme. How should we prove this?

Let p: \mathcal{S}(\mathcal{E}) \to \mathrm{Spec}\,k be the structure map. The identity map \mathcal{S}(\mathcal{E}) \to \mathcal{S}(\mathcal{E}) corresponds to a universal section s^{\mathrm{univ}} \in H^0(\mathcal{S}(\mathcal{E}) \times_k X, (p \times \mathrm{id})^{\ast}\mathcal{E}). Let Z\subset |\mathcal{S}(\mathcal{E}) \times_k X| denote the zero locus of s^{\mathrm{univ}}. This is a closed subset. But now we observe that the projection \pi: \mathcal{S}(\mathcal{E}) \times_k X \to \mathcal{S}(\mathcal{E}) is proper, hence universally closed, and so |\pi|(Z) is a closed subset of |\mathcal{S}(\mathcal{E})|.  One then checks directly that \mathcal{S}(\mathcal{E})^{\times} is the open subscheme corresponding to the open subset |\mathcal{S}(\mathcal{E})| \smallsetminus |\pi|(Z), so we win.

I guess this sort of thing is child’s play for an experienced algebraic geometer, and indeed it took Johan about 0.026 seconds to suggest that one should try to argue using the universal section.  I only cared about the above problem, though, as a toy model for the same question in the setting of a vector bundle \mathcal{E} over a relative Fargues-Fontaine curve \mathcal{X}_S. In this situation, \mathcal{S}(\mathcal{E}) is a diamond over S^\lozenge, cf. Theorem 22.5 here, but it turns out the above argument still works after some minor changes.

Artin vanishing is false in rigid geometry

Sorry for the lack of blogging.  It’s been a busy semester.

Let k be an algebraically closed field, and let X be a d-dimensional affine variety over k.  According to a famous theorem of Artin (Corollaire XIV.3.5 in SGA 4 vol. 3), the etale cohomology groups H^i_{\mathrm{et}}(X,G) vanish for any i > d and any torsion abelian sheaf G on X_{\mathrm{et}}. This is a pretty useful result.

It’s natural to ask if there’s an analogous result in rigid geometry.  More precisely, fix a complete algebraically closed extension k / \mathbf{Q}_p and a d-dimensional affinoid rigid space X=\mathrm{Spa}(A,A^\circ) over k.  Is it true that H^i_{\mathrm{et}}(X,G) vanishes for (say) any i>d and any \mathbf{Z}/n\mathbf{Z}-sheaf G on X_{\mathrm{et}} for n prime to p?

I spent some time trying to prove this before realizing that it fails quite badly.  Indeed, there are already counterexamples in the case where X=\mathrm{Spa}(k \langle T_1,\dots,T_d \rangle,k^\circ \langle T_1, \dots, T_d \rangle) is the d-variable affinoid disk over k.  To make a counterexample in this case, let Y be the interior of the (closed, in the adic world) subset of X defined by the inequalities |T_i| < |p| for all i; more colloquially, Y is just the adic space associated to the open subdisk of (poly)radius 1/p. Let j: Y \to X be the natural inclusion.  I claim that G = j_! \mathbf{Z}/n\mathbf{Z} is then a counterexample.  This follows from the fact that H^i_{\mathrm{et}}(X,G) is naturally isomorphic to H^i_{\mathrm{et},c}(Y,\mathbf{Z}/n\mathbf{Z}), together with the nonvanishing of the latter group in degree i = 2d.

Note that although I formulated this in the language of adic spaces, the sheaf G is overconvergent, and so this example descends to the Berkovich world thanks to the material in Chapter 8 of Huber’s book.

It does seem possible, though, that Artin vanishing might hold in the rigid world if we restrict our attention to sheaves which are Zariski-constructible.  As some (very) weak evidence in this direction, I managed to check that H^2_{\mathrm{et}}(X,\mathbf{Z}/n \mathbf{Z}) vanishes for any one-dimensional affinoid rigid space X.  (This is presumably well-known to experts.)

Riemann-Roch sur la courbe

Let C/\mathbf{Q}_p be a complete algebraically closed extension, and let X = X_{C^\flat} be the Fargues-Fontaine curve associated with C^\flat.  If \mathcal{E} is any vector bundle on X, the cohomology groups H^i(X,\mathcal{E}) vanish for all i>1 and are naturally Banach-Colmez Spaces for i=0,1.  Recall that the latter things are roughly “finite-dimensional C-vector spaces up to finite-dimensional \mathbf{Q}_p-vector spaces”. By a hard and wonderful theorem of Colmez, these Spaces form an abelian category, and they have a well-defined Dimension valued in \mathbf{N} \times \mathbf{Z} which is (componentwise-) additive in short exact sequences.  The Dimension roughly records the C-dimension and the \mathbf{Q}_p-dimension, respectively.  Typical examples are H^0(X, \mathcal{O}(1)) = B_{\mathrm{crys}}^{+,\varphi = p}, which has Dimension (1,1), and H^1(X,\mathcal{O}(-1)) = C/\mathbf{Q}_p, which has Dimension (1,-1).

Here I want to record the following beautiful Riemann-Roch formula.

Theorem. If \mathcal{E} is any vector bundle on X, then \mathrm{Dim}\,H^0(X,\mathcal{E}) - \mathrm{Dim}\,H^1(X,\mathcal{E}) = (\mathrm{deg}(\mathcal{E}), \mathrm{rk}(\mathcal{E})).

One can prove this by induction on the rank of \mathcal{E}, reducing to line bundles; the latter were classified by Fargues-Fontaine, and one concludes by an explicit calculation in that case.  In particular, the proof doesn’t require the full classification of bundles.

So cool!