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Let $X$ be a projective 3-dimensional variety with mild singularities (rational double points). Is there some general result showing that

$$Ext^{1}(\mathcal{F}, \mathcal{O}_{p}) =0$$

where $\mathcal{F}$ is a rank 1, torsion-free coherent sheaf on $X$, $\mathcal{O}_{p}$ is the skyscraper sheaf at a point $p \in X$ (either smooth or singular point), and the Ext group is that of coherent sheaves? I have a very specific $\mathcal{F}$ in mind, but it's extremely difficult to work with explicitly (for context, $X$ is related to a moduli problem and $\mathcal{F}$ is a subsheaf of the universal sheaf).

If there's no such general result, what is the minimum I must dig in to find about $\mathcal{F}$ to show this?

(EDIT: I think the general result is certainly not true after all. As a counterexample, if $p \in X$ is a singular point, then the ideal sheaf of this point $\mathcal{I}_{p}$ is torsion-free, but $Ext^{1}(\mathcal{I}_{p}, \mathcal{O}_{p}) \neq 0$ I believe. So my question becomes...is there some minimal amount I must show about $\mathcal{F}$ for this Ext group to vanish? Or must I just explicitly find resolutions and compute?) .

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The vanishing $Ext^1(\mathcal{F},\mathcal{O}_p) = 0$ is equivalent to local freeness of $\mathcal{F}$ at $p$, see, for example, N. Bourbaki: Éléments de mathématique. Algèbre commutative. Chapitre 10, X.3, Proposition 4.

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  • $\begingroup$ That's excellent, thank you. That fits well with the ideal sheaf counterexample I gave in my edit above. So just to make sure I've got it right: since $\mathcal{F}$ is rank 1, I should be trying to show that the stalk $\mathcal{F}_{p}$ is isomorphic to the local ring $\mathcal{O}_{X, p}$? $\endgroup$
    – Benighted
    Aug 20, 2019 at 21:00
  • $\begingroup$ @Benighted: Correct. $\endgroup$
    – Sasha
    Aug 20, 2019 at 21:22
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    $\begingroup$ Sorry, I hope I don't have a different edition or something, but there doesn't seem to be a Proposition 4 on X.3. Did you mean Proposition 2 perhaps? Sorry, if my French was better I could find this myself! $\endgroup$
    – Benighted
    Aug 20, 2019 at 21:32
  • $\begingroup$ In my pdf file this is page 39. $\endgroup$
    – Sasha
    Aug 20, 2019 at 22:34

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