$\mathrm{\ell }$-adic étale cohomology

Let $X$ be a scheme over a field $k$ having algebraic closure $\overline{k}$. Let ${(X{\otimes }_k\overline{k})}_{\text{ét}}$ be the small étale site on $X{\otimes }_k\overline{k}$, and let $\mathbb{Z}/l^n\mathbb{Z}$ denote the sheaf on ${(X{\otimes }_k\overline{k})}_{\text{ét}}$ associated to the group scheme $\mathbb{Z}/l^n\mathbb{Z}$ for some fixed prime $l$. Finally, let $\mathrm{\Gamma }$ be the global sections functor on the category of étale sheaves on ${(X{\otimes }_k\overline{k})}_{\text{ét}}$.

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

$$H_{\text{ét}}^i(X,{\mathbb{Q}}_l)={\mathbb{Q}}_l{\otimes }_{{\mathbb{Z}}_l}{\underleftarrow{\lim }}_n(R^i\mathrm{\Gamma })(\mathbb{Z}/l^n\mathbb{Z}),.$$

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.

Title	$\mathrm{\ell }$-adic étale cohomology
Canonical name	elladicetaleCohomology
Date of creation	2013-03-22 14:13:39
Last modified on	2013-03-22 14:13:39
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	9
Author	mathcam (2727)
Entry type	Definition
Classification	msc 14F20
Related topic	DerivedFunctor
Related topic	Site
Related topic	EtaleMorphism
Related topic	SmallSiteOnAScheme
Related topic	SheafCohomology