ag.algebraic geometry - Dual of Torsion free/Reflexive Coherent Sheaf

If $E$ is a coherent sheaf on a noetherian scheme, the dual $E^*=Hom_{O_X}(E, O_X)$ is always coherent. If $A$ is an affine open subset, then $E^*$ is the sheaf associated to the $A$-module $Hom_A( Gamma(A, E), Gamma(A, O_X))$. More generally, sheaf hom of any two sheaves preserves coherence.

This is a corollary of the fact that if $M,N$ are finitely presented $A$-modules, then for any multiplicative subset $S$, $S^{-1}Hom_A(M,N) = Hom_{S^{-1}A}(S^{-1}M, S^{-1}N)$, which can be found in any commutative algebra textbook (e.g. Eisenbud).