$Pic(X)/l=0$ in terms of $H^*_{et}(X,\mu_{l^n})$?

Question

I would like to calculate Picard groups of certain schemes over fields; I'm mostly interested in the question whether $Pic(X)$ is infinitely $l$-divisible, i.e. whether $Pic(X)/l=0$, $l$ is a prime distinct from the base field characteristic (the latter could be $0$). I would like to have a characterization of this vanishing in terms of etale cohomology of $l$-torsion sheaves.

Certainly, $Pic(X)\cong H^1_{et}(X,G_m)$; yet this only yields an injection of $Pic(X)/l$ into $H^1(X,\mu_l)$, and I don't know how to control the cokernel.

Upd. So, is there a general method that expresses $Pic(X)/l$ in terms of all of $H^i(X,\mu_{l^n})$ (for $i,n>0$)? What can be said here if $X$ is a variety over an algebraically closed field?

Answer 1

$Pic(X)$ mod $l$ injects into $H^2(X,\mu_l)$ with cokernel the group of elements of order $l$ in the Brauer group of $X$. Depending on your $X$, the Brauer group may be known, or it may be as mysterious as a Tate-Shafarevich group. For example, for a complete smooth surface over a finite field, the Brauer group is conjectured to be finite, but this is not known (it's been proved to be equivalent to the Tate conjecture for the surface).

Answer 2

In characteristic zero, Hodge theory is the best way to approach this. Hodge theory would express $H^2(X,\mathbb \mu_l)=H^2(X,\mathbb Z/l)$ in terms of $H^2(X,\mathbb Z)$. You then use the fact that the classes in $H^2(X,\mathbb Z)$ that come from holomorphic line bundles are exactly Hodge classes, so you try to find the Hodge classes. This works best when $H^2(X,\mathbb Z)$ is nontorsion.

I think that in general $Pic(X)$ being $l$-divisible is rare, e.g. it is never such for any projective variety. One case where it is $l$-divisible is affine curves, where $H^2(\mu_l)$ is zero because $H^2$ of anything locally constant is zero.