Suppose that the stalk of a constructible l-adic ($\mathbb{Z}_l$-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 $S/lS$?
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