ext sheaf and cohomology

Question

Let $\mathcal{F}$ be a sheaf on $\mathbb{P}^{3}$ with $\mbox{dim}(\mathcal{F}) = 0$. It's true that cohomology $H^{i}(\mathbb{P}^{3}, ext^{3}(\mathcal{F}, \mathcal{O}_{\mathbb{P}^{3}})) = 0$ for $i = 1,\, 2,\,3$? If so, what result guarantees this? Thanks in advance.

Answer 1

See Lemma~II.1.1.2 in [Okonek, Christian; Schneider, Michael; Spindler, Heinz. Vector bundles on complex projective spaces].