Comparison results for étale cohomology in rigid geometry

Abstract

We give some comparison results between étale cohomology of schemes and that of associated rigid spaces. First we work in the framework of Fujiwara spaces, and prove a comparison theorem for the derived direct image of morphisms which are not necessarily proper. We need the assumption that the base of the comparison functor consists of schemes of finite type over a field. Next, we consider an analogous problem for adic spaces. In this case, we can prove the comparison theorem under the condition that the base of the comparison functor consists of schemes of finite type over a complete discrete valuation ring with equal characteristic. Finally we propose a definition of the nearby cycle functor for an adic space over a complete discrete valuation ring and give a comparison result on it.