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Let $Y$ an affine finite type scheme over an algebraically closed field $k$. Let $S$ be a closed subscheme of $Y$ and $Y'$ the henselization of $Y$ along $S$. If we have a $\mathbb{Z}_{\ell}$ local system on $Y'$, does it come from a $\mathbb{Z}_{\ell}$ local system on some étale neighbourhood $U\rightarrow Y$ of $S$?

I guess in the case where $S\rightarrow Y$ is a regular immersion it should be true, as for the étale topology I can find a map in the other direction.

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  • $\begingroup$ this is a really interesting question! I think the answer should be no in the case where $Y$ is an elliptic surface minus a section and $S$ is one of the fibers, but I didn't figure out how to prove it. $\endgroup$
    – Will Sawin
    Commented Feb 10, 2016 at 5:15

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