Let $X$ be a scheme over a field $k$ having algebraic closure $\overline{k}$ . Let $(X\otimes_k \overline{k})_{\'et}$ be the small étale site on $X\otimes_k \overline{k}$ , and let $\mathbb{Z}/l^n\mathbb{Z}$ denote the sheaf on $(X\otimes_k \overline{k})_{\'et}$ 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 $(X\otimes_k \overline{k})_{\'et}$ .

The $l$ -adic étale cohomology of $X$ is $$ H^i_\text{\'et}(X,\mathbb{Q}_l) = \mathbb{Q}_l \otimes_{\mathbb{Z}_l} \varprojlim_n (R^i\Gamma)(\mathbb{Z}/l^n\mathbb{Z}),. $$ 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.

Bibliography

1

J. S. Milne, Lectures on Étale Cohomology, 1998, available on the web at 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.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.html