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?

 

2 thoughts on “A counterexample”

  1. The Schloss Elmau counterexample was the following. Take a Tate elliptic curve E with parameter q=p, and consider the \mathbb Q_p-local system L that becomes trivial over the Tate uniformization, and has monodromy p. One can compute the pro-etale H^1 of the rigid-analytic \mathbb G_m with \mathbb Q_p-coefficients by work of Colmez-Niziol, and in particular it contains \mathcal O(\mathbb G_m)/C via Kummer theory. The coordinate function T in there has the property that the monodromy acts via multiplication by q=p, and so this class descends to the cohomology of L on E, as does any C-multiple of it. This gives an embedding of C into H^1(E,L), showing that it is infinite-dimensional.

    I believe the correct version of Kedlaya-Liu’s theorem asserts that one always gets Banach-Colmez spaces.

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