Étale Cohomology of Schemes and Analytic Spaces

Abstract

Etale cohomology was developed in the scheme-theoretic context by Grothendieck in the 50s and 60s in order to provide a purely algebraic cohomology theory, satisfying the same fundamental properties as the singular cohomology of complex varieties, which was needed for proving the Weil conjectures. For other deep arithmetic reasons (related to Langlands program) it appeared later that it should also be worthwhile developing such a theory in the p-adic analytic context. This was done by Berkovich in the early 90s. In this text, after an introduction devoted to the general motivations for building those cohomology theories, we explain what a Grothendieck topology and its associated cohomology theory are. Then we present the basic ideas, definitions, and properties of both scheme-theoretic and Berkovich-theoretic etale cohomology theories (which are closely related to each other), and the fundamental results about them like various GAGA-like comparison theorems and Poincare duality. Our purpose is not to give detailed proofs, which are for most of them highly technical and can be found in the literature. We have chosen to rather insist on examples, trying to show how etale cohomology can at the same time be quite close to the classical topological intuition, and encode in a completely natural manner deep-field arithmetic phenomena (such as Galois theory), which allows sometimes to think to the latter in a purely geometrical way.