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.)