We prove a rigid analytic analogue of the Artin–Grothendieck vanishing theorem. Precisely, we prove (under mild hypotheses) that the geometric étale cohomology of any Zariski-constructible sheaf on any affinoid rigid space $X$ vanishes in all degrees above the dimension of $X$. Along the way, we show that branched covers of normal rigid spaces can often be extended across closed analytic subsets, in analogy with a classical result for complex analytic spaces. We also prove some new comparison theorems relating the étale cohomology of schemes and rigid analytic varieties, and give some applications of them. In particular, we prove a structure theorem for Zariski-constructible sheaves on characteristic-zero affinoid spaces.
Vanishing and comparison theorems in rigid analytic geometry
