Artin vanishing is false in rigid geometry

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

Let be an algebraically closed field, and let be a -dimensional affine variety over .  According to a famous theorem of Artin (Corollaire XIV.3.5 in SGA 4 vol. 3), the etale cohomology groups vanish for any and any torsion abelian sheaf on . 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 and a -dimensional affinoid rigid space over .  Is it true that vanishes for (say) any and any -sheaf on for  prime to ?

I spent some time trying to prove this before realizing that it fails quite badly.  Indeed, there are already counterexamples in the case where  is the -variable affinoid disk over .  To make a counterexample in this case, let be the interior of the (closed, in the adic world) subset of defined by the inequalities for all ; more colloquially,  is just the adic space associated to the open subdisk of (poly)radius . Let be the natural inclusion.  I claim that is then a counterexample.  This follows from the fact that is naturally isomorphic to , together with the nonvanishing of the latter group in degree .

Note that although I formulated this in the language of adic spaces, the sheaf 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 vanishes for any one-dimensional affinoid rigid space .  (This is presumably well-known to experts.)

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