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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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Yes, because any constructible $\ell$-adic sheaf on a noetherian scheme is lisse on the constituents of a stratification. –  BCnrd Nov 9 '10 at 20:06
Sorry; could you also tell me why the fact is true for lisse sheaves? –  Mikhail Bondarko Nov 9 '10 at 21:32
Dear Mikhail: I'll give a reference for this important fact: Prop. 12.10 in Chapter I of Freitag-Kiehl (where their scheme $X$ is assumed to be noetherian). They have a small typo: $\mathcal{O}_n$ should be $\mathcal{C}_n$ on line -2 of that proof. –  BCnrd Nov 10 '10 at 1:19
Another reference would be SGA 4 1/2, Rapport, 2.5. –  shenghao Feb 3 '11 at 22:13
Thank you!!!!!! –  Mikhail Bondarko Feb 3 '13 at 6:25

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