nLab ℓ-adic cohomology

Idea

$\ell$-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.

Definition

Over the étale site

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

$$H^j_{et}(X_{\overline{k}}, \mathbb{Q}_\ell) \coloneqq \lim_n H^j_{et}(X_{\overline{k}}, \mathbb{Z}/\ell^n\mathbb{Z})\otimes_{\mathbb{Z}_{\ell}} \mathbb{Q}_\ell$$

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

The key insights into getting finite dimensionality with coefficients in a field of characteristic $0$ when $k$ has positive characteristic is to first base change to $\overline{k}$ 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)

Related concepts

• Weil cohomology

  • étale cohomology, basics of étale cohomology

  • pro-étale cohomology

• p-adic integers

• p-adic geometry

• p-adic homotopy

• p-adic cohomology

• p-adic physics

References

A textbook account is in

• James Milne, section 19 of Lectures on Étale Cohomology (web)

Surveys include

• Akhil Mathew, $l$-adic cohomology and exponential sums (web)

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

• Bhargav Bhatt, Peter Scholze, The pro-étale topology for schemes (arXiv:1309.1198)