Does there exist any result that characterize reflexive sheaves by the annulment of ext sheaf?

references on this subject are welcome!

Thanks a lot.

  • $\begingroup$ What is meant by the annulment of a sheaf? $\endgroup$ – mathphys Jul 4 '19 at 16:19
  • $\begingroup$ for example, Hartshorne (Algebraic Geometry) in the exercise 6.5, Cap III, says that $\mbox{ext}^{i}(F, G) = 0 \Leftrightarrow $ $F$ is locally free. It's this kind of nullification I'm looking for. $\endgroup$ – Allan Ramos Jul 4 '19 at 17:24

Here's a result that links reflexive sheaves with the codimension of related ext sheaves, from page 6 of The Geometry of Moduli Spaces of Sheaves by Huybrechts and Lehn:

Let $\mathcal{F}$ be a coherent sheaf of dimension $c$ on a smooth projective variety $X$. Then $\mathcal{F}$ is a reflexive sheaf if and only if $\mathrm{codim}(\mathcal{E}xt^q(\mathcal{F}, \omega_X)) \geq q + 2 $ for all $q > c$

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  • $\begingroup$ What definition of reflexive are you using? If $c<\dim X$, then such a sheaf can not be reflexive over $X$ since dual has to be zero, so you meant something else. $\endgroup$ – Mohan Jul 4 '19 at 18:08
  • $\begingroup$ @mathphys. All right, I got it. If $\mbox{dim}X = n$ , then $c = n - d$, where $d = \mbox{dim}E$. Thank you very much. $\endgroup$ – Allan Ramos Jul 4 '19 at 20:10

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