nLab ℓ-adic cohomology





Special and general types

Special notions


Extra structure



Étale morphisms



\ell-Adic cohomology is a cohomology theory on suitable varieties which is constructed as an inverse limit of étale cohomology over different coefficients.

\ell-Adic cohomology is a Weil cohomology theory. It is endowed with a Galois action and therefore gives us a way of obtaining Galois modules from algebraic varieties.


Over the étale site

Let XX be a (smooth?) proper variety over a field. Fix \ell a prime number different from the characteristic of kk. The \ell-adic cohomology is defined to be the cohomology theory on the étale site given by the inverse limit over nn \in \mathbb{N}

H et j(X k¯, )lim nH et j(X k¯,/ n) H^j_{et}(X_{\overline{k}}, \mathbb{Q}_\ell) \coloneqq \lim_n H^j_{et}(X_{\overline{k}}, \mathbb{Z}/\ell^n\mathbb{Z})\otimes_{\mathbb{Z}_{\ell}} \mathbb{Q}_\ell

of étale cohomology with coefficients in / n\mathbb{Z}/\ell^n\mathbb{Z}.

The key insights into getting finite dimensionality with coefficients in a field of characteristic 00 when kk has positive characteristic is to first base change to k¯\overline{k} to make the theory “geometric.” Then only work with torsion sheaves so that appropriate finiteness theorems for étale cohomology of proper varieties can be used, and then pass to the limit.

Notice that on the left \mathbb{Q}_{\ell} is not an actual sheaf whose actual sheaf cohomology is being computed instead the expression on the left is defined by the genuine sheaf cohomology groups on the right.

Over the pro-étale site

This is rectified by passing to pro-étale cohomology. Here \mathbb{Q}_{\ell} exists as an actual sheaf and its genuine abelian sheaf cohomology is \ell-adic cohomology. (Bhatt-Scholze 13, section 6.8)


A textbook account is in

Surveys include

A variant of the étale site, well adapted to the needs of \ell-adic cohomology, the pro-étale site (locally contractible in some sense) is discussed in

Last revised on July 2, 2022 at 18:56:16. See the history of this page for a list of all contributions to it.