# Artin vanishing is false in rigid geometry

Sorry for the lack of blogging.  It’s been a busy semester.

Let $k$ be an algebraically closed field, and let $X$ be a $d$-dimensional affine variety over $k$.  According to a famous theorem of Artin (Corollaire XIV.3.5 in SGA 4 vol. 3), the etale cohomology groups $H^i_{\mathrm{et}}(X,G)$ vanish for any $i > d$ and any torsion abelian sheaf $G$ on $X_{\mathrm{et}}$. This is a pretty useful result.

It’s natural to ask if there’s an analogous result in rigid geometry.  More precisely, fix a complete algebraically closed extension $k / \mathbf{Q}_p$ and a $d$-dimensional affinoid rigid space $X=\mathrm{Spa}(A,A^\circ)$ over $k$.  Is it true that $H^i_{\mathrm{et}}(X,G)$ vanishes for (say) any $i>d$ and any $\mathbf{Z}/n\mathbf{Z}$-sheaf $G$ on $X_{\mathrm{et}}$ for $n$ prime to $p$?

I spent some time trying to prove this before realizing that it fails quite badly.  Indeed, there are already counterexamples in the case where $X=\mathrm{Spa}(k \langle T_1,\dots,T_d \rangle,k^\circ \langle T_1, \dots, T_d \rangle)$ is the $d$-variable affinoid disk over $k$.  To make a counterexample in this case, let $Y$ be the interior of the (closed, in the adic world) subset of $X$ defined by the inequalities $|T_i| < |p|$ for all $i$; more colloquially, $Y$ is just the adic space associated to the open subdisk of (poly)radius $1/p$. Let $j: Y \to X$ be the natural inclusion.  I claim that $G = j_! \mathbf{Z}/n\mathbf{Z}$ is then a counterexample.  This follows from the fact that $H^i_{\mathrm{et}}(X,G)$ is naturally isomorphic to $H^i_{\mathrm{et},c}(Y,\mathbf{Z}/n\mathbf{Z})$, together with the nonvanishing of the latter group in degree $i = 2d$.

Note that although I formulated this in the language of adic spaces, the sheaf $G$ is overconvergent, and so this example descends to the Berkovich world thanks to the material in Chapter 8 of Huber’s book.

It does seem possible, though, that Artin vanishing might hold in the rigid world if we restrict our attention to sheaves which are Zariski-constructible.  As some (very) weak evidence in this direction, I managed to check that $H^2_{\mathrm{et}}(X,\mathbf{Z}/n \mathbf{Z})$ vanishes for any one-dimensional affinoid rigid space $X$.  (This is presumably well-known to experts.)