# When is the etale cohomology of a surface free?

Let $X$ be a smooth variety over $\mathbb{F}_q$, $\ell$ a prime prime to $q$. I learned that if $X$ is a curve then $H_c^i(X,\mathbb{Z}_{\ell})$ is a free $\mathbb{Z}_{\ell}$-module.

When $X$ is of a higher dimension, is there some general condition that can be added to $X$ so that the same conclusion holds? (I am primarily interested in the case that $X$ is a surface.)

• Note that the answer for general $X$ will depend on $\ell$. In fact, Gabber proved that is $X$ is smooth projective, then for almost all $\ell$ the cohomology $H^i(X,\mathbb Z_\ell)$ is torsion free. (In characteristic $0$ this easily follows from comparison with singular cohomology; the proof in characteristic $p$ uses the deep 'gcd theorem' [Weil II, Thm. 4.5.1] from Deligne's work on the Weil conjectures.) It seems likely that this implies a similar statement for $X$ smooth quasi-projective. – R. van Dobben de Bruyn Jan 11 '18 at 4:12
• Thus, if you want to know torsion-freeness for all $\ell$, this is morally equivalent to the question whether the singular cohomology of a smooth (projective) complex surface is torsion free. There is probably no easy condition on the surface that implies this. – R. van Dobben de Bruyn Jan 11 '18 at 4:22
• Note that in char 0, $H^3_{tors} = H^2_{tors} = (H_1)_{tors} = (\pi_1^{ab})_{tors}$. An example with torsion is an Enriques surface, where $\pi_1 = \mathbb{Z}/2$ – Kevin Casto Jan 11 '18 at 5:19
• @R.vanDobbendeBruyn This sounds exactly what I need! Could you elaborate more details on the quasi-projective case? – user148212 Jan 11 '18 at 5:40
• The Gabber reference is Sur la torsion dans la cohomologie $\ell$-adique d’une variété, C. R. Acad. Sci., Paris, Sér. I 297(1) (1983), p. 179-182. It is available online through the Bibliothèque nationale de France. – R. van Dobben de Bruyn Jan 11 '18 at 6:48