# Reflexive sheaf and ext sheaf

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

references on this subject are welcome!

Thanks a lot.

• What is meant by the annulment of a sheaf? – mathphys Jul 4 '19 at 16:19
• 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. – Allan Ramos Jul 4 '19 at 17:24

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 $$q > c$$
• 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. – Mohan Jul 4 '19 at 18:08
• @mathphys. All right, I got it. If $\mbox{dim}X = n$ , then $c = n - d$, where $d = \mbox{dim}E$. Thank you very much. – Allan Ramos Jul 4 '19 at 20:10