A counterexample


Let $latex C/mathbf{Q}_p$ be a complete algebraically closed nonarchimedean field extension, and let $latex X$ be any proper rigid space over $latex C$. Let $latex mathbf{L}$ be any $latex mathbf{Z}_p$-local system on $latex X_{mathrm{proet}}$. By the main results in Scholze’s p-adic Hodge theory paper, the pro-etale cohomology groups $latex H^i_{mathrm{proet}}(X,mathbf{L})$ are always finitely generated $latex mathbf{Z}_p$-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 $latex mathbf{Q}_p$-local system $latex mathbf{V}$. By analogy, one might guess that the cohomology groups $latex H^i_{mathrm{proet}}(X,mathbf{V})$ are always finitely generated $latex mathbf{Q}_p$-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 $latex X=mathbf{P}^1$ as a rigid space over $latex C$. This is the target of the…

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