We prove a rigid analytic analogue of the Artin–Grothendieck vanishing theorem. Precisely, we prove (under mild hypotheses) that the geometric etale cohomology of any Zariski-constructible sheaf on any affinoid rigid space vanishes in all degrees above the dimension of . 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 etale 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.

中文翻译：

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刚性解析几何中的消失和比较定理

我们证明了 Artin-Grothendieck 消失定理的严格解析模拟。准确地说，我们证明（在温和的假设下）任何 Zariski 可构造层在任何仿射刚性空间上的几何 etale 上同调在 的维数以上的所有度数上都消失。在此过程中，我们证明了正常刚性空间的分支覆盖通常可以扩展到封闭的解析子集，类似于复杂解析空间的经典结果。我们还证明了一些新的比较定理，这些定理涉及方案的永恒上同调和刚性分析簇，并给出了它们的一些应用。特别是，我们证明了特征零仿射空间上 Zariski 可构造滑轮的结构定理。