étale $\ell$-adic cohomology is a Weil cohomology theory

Question

I was reading https://mathoverflow.net/questions/85078/ell-adic-weil-cohomology-theory and in the first paragraph it is said that

$\ell$-adic cohomology is a Weil cohomology theory over separably closed fields.

I have been looking for a reference for that everywhere, but I haven't found any precise statement. Could someone help me?

For Weil cohomology theory I stick to the definition given in Stacks Project https://stacks.math.columbia.edu/tag/0FHY.

In general, I would like to know the precise hypotheses on the base field for $\ell$-adic cohomology to be a Weil cohomology theory:

For example: is $\ell$-adic cohomology a Weil cohomology theory on the category of smooth projective schemes over $k$, when $k$ is

1. a separably closed field
2. a finite field
3. a characteristic zero field (not necessarily algebraically closed)
4. an algebraically closed field

My guesses are:

1. should be true
2. is false, and a counterexample is in Remark 17.9 in Milne's Lecures on étale cohomology.
3. is flase, in the third row of https://link.springer.com/article/10.1007/BF01456052 it is said that over number fields the cohomology groups need not be finite dimensional. But again, I couldn't find any reference with a proof.
4. should be true

I would like to find some official references, with proofs.

Thanks in advance to those who can answer me.

Answer 1

Let $k$ be a base field and let $H^*(-) \colon SmProj/k \longrightarrow \left \{\text{graded} \ K \ \text{algebras} \right\}$ for some field $K$ of characteristic zero functor. In order to be a Weil cohomology theory, it should satisfy some axioms below

• $H^i(X)$ is finitely dimensional $K$-vector space for all $i$.
• $H^i(X) = 0$ for any $i< 0$ or $i > 2\dim(X)$.
• $H^{2\dim(X)}(X) \simeq K$ and there is a perfect pairing $H^i(X) \times H^{2\dim(X)-i}(X) \longrightarrow H^{2\dim}(X) \simeq K$.
• Kunneth formula holds $H^*(X) \otimes H^*(Y) \simeq H^*(X \times Y)$.
• cycles class maps, weak + hard Lefschetz theorems, lifting to $\mathbb{C}$,... that I prefer to avoid here.

$l$-adic cohomology is such a cohomology theory and it can be defined over any field $k$ of characteristic different from $l$. There is no need to restrict to smooth projective $k$-varieties and we can start with $k$-varieties from the beginning. We simply set $$H^i(X) = H^i_{et}(X_{\overline{k}},\mathbb{Q}_l)= \varprojlim H^i_{et}(X_{\overline{k}},\mathbb{Z}/l^n\mathbb{Z}) \otimes_{\mathbb{Z}_l} \mathbb{Q}_l$$ (here $K = \mathbb{Q}_l$) and the reason that you have to base change to $\overline{k}$ is simply because it work best when the base field is algebraically closed (in fact, separably closed is enough). I may assume from now that $k=\overline{k},X=X_{\overline{k}}$ to avoid cumbersome notation.

I explain below some places where working with algebraically closed is essential. I guess that your problem with textbooks in etale cohomology is that you seem to not be aware that you should base change to algebraic closure.

• The finite dimension property of $H^i_{et}(X,\mathcal{F})$ for $\mathcal{F}$ torsion is what you have read in Milne's note, it is not finite if the base field is not algebraically closed.
• The vanishing of $H^i_{et}(X,\mathcal{F})$ when $i>2\dim(X)$ for $\mathcal{F}$ torsion (see Theorem 7.5.5 in Leifu) is a corollary of bounding Galois cohomological dimension (see Theorem 4.5.9 + 4.5.10 in Leifu).
• For duality, you need to build duality for curves first. That is done by knowing the explicit cohomology groups of a curve and it boils down to Tsen's theorem (see Theorem 7.2.9 in Leifu or Proposition 5.1 in Freitag), i.e. $H^{i>0}(K,\mathbb{G}_m)=0$ for $K$ a function field in one variable over an algebraically closed field.
• Kunneth formula is a consequence of the projection formula and the proper base change theorem. In fact, in the proof of the projection formula you need the proper base change theorem to reduce your work to the spectrum of a separably closed field over which sheaves are constant (see Proposition 8.14 in Freitag).