Exercise 7, § 4

Let $A$ be an integrally closed Noetherian domain, $E$ a finitely generated torsion-free $A$-module and $E^*$ its dual. (a) Show that the canonical homomorphism $E^* \otimes_A E \to \operatorname{End}_A(E)$ is a pseudo-isomorphism. (b) Deduce from (a) that for $E$ to be a projective $A$-module, it is necessary and sufficient that $E^* \otimes_A E$ be a reflexive $A$-module. (Note that, if $E^* \otimes_A E$ is reflexive, the canonical homomorphism $E^* \otimes_A E \to \operatorname{End}_A(E)$ is bijective.)