Suppose that the stalk of a constructible ladic ($\mathbb{Z}_l$adic or $\mathbb{Q}_l$adic) etale sheaf $S$ over a generic (Zariski) point of a variety $V$ is zero. Does this imply that $S$ vanishes over an open subvariety of $V$ (containing this generic point)? This is obviously wrong without constructibilty, and obviously true for torsion sheaves; it is not quite clear for me what happens for $l$adic sheaves. Is it sufficient to consider $S/lS$?
