Artin vanishing in rigid analytic geometry

TL;DR

This paper establishes a vanishing theorem for geometric etale cohomology in rigid analytic geometry, extending classical results and providing comparison theorems between algebraic and analytic cohomologies.

Contribution

It proves a rigid analytic analogue of Artin vanishing, extending classical theorems to the rigid analytic setting and relating algebraic and analytic cohomologies.

Findings

Vanishing of higher cohomology for Zariski-constructible sheaves on affinoid spaces

Extension of branched covers across closed analytic subsets

Comparison theorem between algebraic and analytic etale cohomologies

Abstract

We prove a rigid analytic analogue of the Artin vanishing theorem. Precisely, we prove (under mild hypotheses) that the geometric etale 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 a general comparison theorem relating the algebraic and analytic etale cohomologies of any affinoid rigid space.