Comparison theorems

Abstract

In this chapter we will explain some comparison theorems. Our first and most important theorem is stated in §2. It reduces the computation of the étale cohomology of certain subsets of affinoid adic spaces to the computation of the étale cohomology of affine schemes. This theorem is the basic ingredient for the proof of many results of this book. In §5 we will study the relation between the étale cohomology of a formal scheme X and the étale cohomology of its associated rigid analytic variety X^an. For example, we will prove a theorem which concerns the stalks of the sheaves Rⁿ λ_* (F)where λ: (X^an)_ét → X_ét is the natural morphism of sites and F is a sheaf on (X^an)_ét. In later chapters we will apply this theorem to the étale cohomology with proper support for morphisms of rigid analytic varieties.