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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asked Nov 9, 2010 at 19:44
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2$\begingroup$ Yes, because any constructible $\ell$-adic sheaf on a noetherian scheme is lisse on the constituents of a stratification. $\endgroup$– BCnrdCommented Nov 9, 2010 at 20:06
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$\begingroup$ Sorry; could you also tell me why the fact is true for lisse sheaves? $\endgroup$– Mikhail BondarkoCommented Nov 9, 2010 at 21:32
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2$\begingroup$ 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. $\endgroup$– BCnrdCommented Nov 10, 2010 at 1:19
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$\begingroup$ Another reference would be SGA 4 1/2, Rapport, 2.5. $\endgroup$– shenghaoCommented Feb 3, 2011 at 22:13
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