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.)