Abstract
Étale 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 étale cohomology theories (which are closely related to each other), and the fundamental results about them like various GAGA-like comparison theorems and Poincaré 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 étale 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.
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Notes
- 1.
It can be shown that X satisfies this property if and only if it is supersingular, that is, if and only if its p-torsion subgroup consists of one \(\overline{\mathbb{F}_{p}}\)-point, with multiplicity p 2.
- 2.
There could be one because \(\mathcal{F}(X) \rightarrow \mathcal{F}(Y )\) not only depends on Y, but also on the arrow Y → X.
- 3.
The latter category is in fact (equivalent to) the category of sheaves of abelian groups on a suitable site.
- 4.
- 5.
This result was written in the rigid-analytic language and was actually the first motivation for developing such a theory.
- 6.
This theorem asserts that if \(L/k\) is a finite Galois extension with cyclic Galois group \(\langle \sigma \rangle\), every element of L whose norm is equal to one can be written σ(x)∕x for some x ∈ L ∗.
- 7.
This can also be expressed as a comparison between étale and Zariski \(\mathrm{H}^{1}\), but with coefficients in GL n —which is not an abelian group.
- 8.
If X is only assumed to be smooth, this is still possible using Grothendieck’s GAGA results for de Rham cohomology, which involve resolution of singularities.
- 9.
The fact for the action of G on the cohomology of \(\mathcal{F}_{\widehat{k^{a}}}\) to be discrete essentially means that every cohomology class of \(\mathcal{F}_{\widehat{k^{a}}}\) is already defined on a finite separable extension of k; one needs compactness to ensure this finiteness result.
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Acknowledgements
I would like to thank warmly: Mathilde Herblot, who has written detailed notes during the lectures, and has allowed me to use them for the redaction of this text; and Johannes Nicaise, who has read carefully several versions of this work and has made a lot of remarks and suggestions, which greatly helped me to improve the initial presentation.
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Ducros, A. (2015). Étale Cohomology of Schemes and Analytic Spaces. In: Ducros, A., Favre, C., Nicaise, J. (eds) Berkovich Spaces and Applications. Lecture Notes in Mathematics, vol 2119. Springer, Cham. https://doi.org/10.1007/978-3-319-11029-5_2
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