[Math] Dual of Torsion free/Reflexive Coherent Sheaf

Question

If $E$ is a coherent sheaf on projective space then is it always true that its dual $ E^* $ is also coherent.? Moreover, if $E$ is also Torsion free or Reflexive then, $E^* $ is also Torsion free or Reflexive?

Answer 1

The dual of any coherent sheaf on a projective space is reflexive (and in particular torsion free). The easiest way to see it is the following. Choose a locally free resolution $\dots \to F_1 \to F_0 \to E \to 0$. After dualizing it gives an exact sequence $0 \to E^* \to F_0^* \to F_1^*$, thus $E^*$ is a kernel of a morphism of locally free sheaves, hence reflexive.