$\ell$-adic étale cohomology

Let $X$ be a scheme over a field $k$ having algebraic closure $\overline{k}$. Let $(X\otimes_{k}\overline{k})_{\text{\'{e}t}}$ be the small étale site on $X\otimes_{k}\overline{k}$, and let $\mathbb{Z}/l^{n}\mathbb{Z}$ denote the sheaf on $(X\otimes_{k}\overline{k})_{\text{\'{e}t}}$ associated to the group scheme $\mathbb{Z}/l^{n}\mathbb{Z}$ for some fixed prime $l$. Finally, let $\Gamma$ be the global sections functor on the category of étale sheaves on $(X\otimes_{k}\overline{k})_{\text{\'{e}t}}$.

The $l$-adic étale cohomology of $X$ is

 $H^{i}_{\text{\'{e}t}}(X,\mathbb{Q}_{l})=\mathbb{Q}_{l}\otimes_{\mathbb{Z}_{l}}% \varprojlim_{n}(R^{i}\Gamma)(\mathbb{Z}/l^{n}\mathbb{Z}),.$

where $R^{i}$ denotes taking the $i$-th right-derived functor.

This apparently appalling definition is necessary to ensure that (for $l$ not equal to the characteristic of $k$) étale cohomology is the appropriate generalization of de Rham cohomology on a complex manifold. For example, on a scheme of dimension $n$, the cohomology groups $H^{i}$ vanish for $i>2n$ and we have a version of Poincaré duality. Grothendieck introduced étale cohomology as a tool to prove the Weil conjectures, and indeed it is what Deligne used to prove them.

These references are approximately in order of difficulty and of generality and precision.

References

• 1 J. S. Milne, Lectures on Étale Cohomology, 1998, available on the web at http://www.jmilne.org/math/http://www.jmilne.org/math/
• 2 James S. Milne, Étale cohomology, volume 33 of Princeton Mathematical Series. Princeton University Press, Princeton N.J., 1980
• 3 Deligne et al., Séminaires en Gèometrie Algèbrique 4$\frac{1}{2}$, available on the web at http://www.math.mcgill.ca/ archibal/SGA/SGA.htmlhttp://www.math.mcgill.ca/ archibal/SGA/SGA.html
• 4 Grothendieck et al., Séminaires en Gèometrie Algèbrique 4, tomes 1, 2, and 3, available on the web at http://www.math.mcgill.ca/ archibal/SGA/SGA.htmlhttp://www.math.mcgill.ca/ archibal/SGA/SGA.html
Title $\ell$-adic étale cohomology elladicetaleCohomology 2013-03-22 14:13:39 2013-03-22 14:13:39 mathcam (2727) mathcam (2727) 9 mathcam (2727) Definition msc 14F20 DerivedFunctor Site EtaleMorphism SmallSiteOnAScheme SheafCohomology