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I've asked this question here but never got an answer, a simplified version of the question is the following:

Given an intersection of hypersurfaces $Y$ on a smooth projective variety $X$ over a finite field, is the category of vector bundles defined on a neighborhood of the $Y$ equivalent to the category of vector bundles on the formal neighborhood of $Y$ ($\text{dim}Y\geq 2$)? This is asking whether $\text{Leff(Y,X)}$ holds or not (This is true if we work over an algebraically closed field)

This seems to be true in the case of very ample divisors (one hypersurface). This example claims it to be true without assumptions on the field. The example does not claim the extension is unique on the neighborhood, so I am wondering whether an ample divisor $Y$ in a smooth projective variety $X$ of dimension $\geq 3$ satisfies $\text{Leff}(Y,X)$ or not? (field is not algebraically closed)

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    $\begingroup$ The cited example refers to this proposition, which comes with a proof. So, what is the question exactly? $\endgroup$ Feb 28, 2021 at 7:48
  • $\begingroup$ So the question is whether Hartshorne's theorem of $Leff(Y,X)$ holds over non-algebraically closed fields like finite fields or not. Another one is whether an ample divisor has the property $Leff(Y,X)$ or not over arbitrary field. The example claims there is an extension from formal neighborhood to a neighborhood but does not claim it is unique. $\endgroup$
    – user127776
    Feb 28, 2021 at 17:09
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    $\begingroup$ Over arbitrary fields, for the hypersurface case, Lef(Y, X) is stacks.math.columbia.edu/tag/0EL2 and Leff(Y, X) is stacks.math.columbia.edu/tag/0EL7. This trivially implies the result for a complete intersection by composing restriction functors several times. $\endgroup$
    – Johan
    Mar 1, 2021 at 0:59

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