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?
2 thoughts on “A counterexample”
Reblogged this on mayorliatmath.
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.