# Vanishing Ext Group of Torsion-free and Skyscraper Sheaves?

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?) .

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.
• 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 $\mathcal{O}_{X, p}$? Aug 20, 2019 at 21:00