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Let $\mathscr{F}$ be a coherent sheaf on a scheme $X$ with reasonable assumptions. Obviously if I restrict to any point $x \in X$, the restriction $\mathscr{F}|_x$ is free over $x$.

I am interested in a the following question: I fix a natural number $n$ and I consider all subschemes $T \subseteq X$ which are thickenings of points of at most order $n$. I now want to describe those coherent sheaves $\mathscr{F}$ on $X$ such that when restricted to any such $T$ are free (or equivalently locally free).

If $n = 1$, then all coherent sheaves have this property. If it holds for all $n$, then I believe it implies that $\mathscr{F}$ is locally free since we can check this on a formal neighbourhood. What about for a fixed $n$?

Could this be connected to the notion of depth of a sheaf?

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    $\begingroup$ A possible source of lots of such sheaves: if $X$ is over $\mathbf{F}_p$ and $F\colon X\to X$ is the Frobenius, then for any given $n\geq 0$ there is an $e\geq 0$ such that for every coherent sheaf $\mathcal{F}$ on $X$, the $e$-th pullback $(F^e)^* \mathcal{F}$ has this property. $\endgroup$
    – Piotr Achinger
    Commented Apr 18 at 20:09
  • $\begingroup$ Do you mean that for every $n$, there exists an $e$ such that the property holds? $\endgroup$
    – ofiz
    Commented Apr 19 at 8:39

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Let $X$ be a nice enough scheme of positive dimension, and let $D$ be a divisor on $X$ which defines some ideal sheaf $I$. Then $\mathcal O_X / I^n$ is not locally free but the restriction of $\mathcal O_X / I^n$ to any thickening of a point of order at most $n$ is free, since if the point does not lie in $D$ the restriction is trivial and if the point lies in $D$ the restriction is free since $I$ lies in the ideal of the point so $I^n$ lies in the $n$th power of the ideal.

(Idea inspired by Piotr Achinger's comment but modified to work in all characteristics.)

All of these schemes for $n\geq 2 $ should have the same depth, so it's not clear how depth should be relevant here, unless some positive result is true for sheaves of very large depth (which I don't think is true either).

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  • $\begingroup$ This makes sense. Perhaps what I would like to know is the converse: if depth of a sheaf $\mathscr{F}$ is greater than $n+1$, is it true that the property holds for all thickenings $T$ of order less than or equal to $n$? In particular $n=1$. $\endgroup$
    – ofiz
    Commented Apr 19 at 12:30
  • $\begingroup$ @ofiz The structure sheaf of a smooth divisor in a smooth variety has depth equal to the dimension of the divisor, i.e. arbitarily large, but the property fails for $n=2$. $\endgroup$
    – Will Sawin
    Commented Apr 19 at 12:54

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