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.)

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. $\endgroup$ – R. van Dobben de Bruyn Jan 11 '18 at 4:22Sur la torsion dans la cohomologie $\ell$-adique d’une variété,C. R. Acad. Sci., Paris, Sér. I297(1)(1983), p. 179-182. It is available online through the Bibliothèque nationale de France. $\endgroup$ – R. van Dobben de Bruyn Jan 11 '18 at 6:48