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.