Let be a complete algebraically closed nonarchimedean field extension, and let be any proper rigid space over . Let be any -local system on . By the main results in Scholze’s p-adic Hodge theory paper, the pro-etale cohomology groups are always finitely generated -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 -local system . By analogy, one might guess that the cohomology groups are always finitely generated -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 as a rigid space over . This is the target of the Gross-Hopkins period map , where is (the rigid generic fiber of the base change to of) the Lubin-Tate deformation space of some fixed connected p-divisible group of dimension 1 and height 2. The rational p-adic Tate module of the universal p-divisible group descends along to a rank two local system on .

**Theorem. ***Maintain the above setup. Then*

i. For any the group is zero.

*ii. The group is a Banach-Colmez space over of Dimension .*

*iii. The group is a Banach-Colmez space over of Dimension .*

Recall that a Banach-Colmez space is a special kind of topological -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 -vector space defined up to a finite-dimensional -vector space. In particular, any such space has a well-defined Dimension, which is a pair in whose entries record the -dimension and the -dimension of the space, respectively. So for example the space has Dimension , and the space has Dimension . Unsurprisingly, any Banach-Colmez space whose -dimension is positive will be disgustingly infinitely generated as a -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 be the evident sheaf on , where e.g. 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

of sheaves on . 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 in this setup as the period domain parametrizing admissible length one modifications of the bundle on the Fargues-Fontaine curve. (Nb. The mysterious middle term in the sequence is really the sheaf of -equivariant maps from to .)

Anyway, this reduces us to computing the groups and . 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 be any Banach-Colmez space over . For any proper rigid space , we may regard as a (pre)sheaf on , so in particular we can talk about the pro-etale cohomology groups . In this notation, the natural map 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 , where it’s a tautology, and the space Colmez notates , 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 . By the standard easy computation of , we get that is a copy of for either of , and it vanishes otherwise. In particular, in degrees 0 and 2, it has Dimension by some of the calculations in Colmez’s original paper.

Next, we need to compute the pro-etale cohomology of . For this, we use the fact (already in Gross and Hopkins’s original article) that . Let be the evident projection of sites. Combining the description of with an easy projection formula gives an isomorphism . Moreover, identifies with in degree zero and in degree 1, and vanishes otherwise. In particular, the only nonvanishing cohomology sheaves of are in degree 0 and in degree 1; moreover, the latter has no global cohomology in any degree. Feeding this information into the Leray spectral sequence for , we get that , so this is for and zero otherwise.

Finally, we can put everything together and take the long exact sequence in pro-etale cohomology associated with . It’s easy to check that doesn’t have any global sections, and the middle term of has no cohomology in degree one, so we get a short exact sequence . We’ve already identified the here as something of Dimension , so by the additivity of Dimensions in short exact sequences, we deduce that has Dimension , as desired. By a similar argument, we get an isomorphism , which we already observed has Dimension . The vanishing of all the other cohomologies of also follows easily.

BTW, there is nothing special about height in this story; I just stuck with it for convenience. For any heght , there is an analogous rank -local system on , and one can check that e.g. has Dimension .

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 -local systems without globally defined lattices, and he also suggested that the cohomologies of 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 as a possible counterexample. Immediately after I gave a lecture on the Tan-Tong paper the next day, the sequence 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 -local systems on proper rigid spaces always Banach-Colmez spaces?

Reblogged this on mayorliatmath.

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The Schloss Elmau counterexample was the following. Take a Tate elliptic curve with parameter , and consider the -local system that becomes trivial over the Tate uniformization, and has monodromy . One can compute the pro-etale of the rigid-analytic with -coefficients by work of Colmez-Niziol, and in particular it contains via Kummer theory. The coordinate function in there has the property that the monodromy acts via multiplication by , and so this class descends to the cohomology of on , as does any -multiple of it. This gives an embedding of into , 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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