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?



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


