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?

 

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.