Summer School of Mathematics "Berkovich Spaces"
Étale Cohomology (Lecture 4) Antoine Ducros (Paris VI) 9 juillet 2010 
Étale cohomology was introduced in the schemetheoretic context by Grothendieck in the 50’s and 60’s 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 padic analytic context. This was done by Berkovich in the early 90’s. In this series of lectures, I plan, after having given some general motivations, to spend some time about the notion of a Grothendieck topology and its associated cohomology theory. Then I will explain the basic ideas and properties of both schemetheoretic and Berkovichtheoretic étale cohomology theories (which are closely related to each other), and the fundamental results like various comparison theorems, Poincaré duality, purity and so forth. My purpose is not to give detailed proofs, which are for most of them highly technical. I will rather insist on examples, trying to show how étale cohomology can at the same time be quite close to the classical topological intuition, and deal in a completely natural manner with deepfield arithmetic phenomena (such as Galois theory), which allows sometimes to think to the latter in a purely geometrical way.
– Lecture 1: July 5, 09:00
– Lecture 2: July 6, 11:30
– Lecture 3: July 7, 10:30
– Lecture 4: July 9, 09:00
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Antoine Ducros (Paris VI) Antoine Ducros est professeur à l’université Paris VI (Institut mathématique de Jussieu), responsable du projet "Espaces de Berkovich" de l’Agence nationale de la recherche. 