Proposition 22

Let $A$ be a Dedekind domain and $M$ a finitely generated $A$-module. The following properties are equivalent: (a) $M$ is torsion-free. (b) $M$ is reflexive. (c) $M$ is projective.

We already know (with no hypothesis on the integral domain $A$) that (b) implies (a) (no. 2, Remark 1) and that (c) implies (b) (Algebra, Chapter II, § 2, no. 7, Corollary 4 to Proposition 14). If $M$ is torsion-free, it is identified with a lattice of $V = M \otimes_A K$ with respect to $A$; $M$, is therefore a free $A_p$-module for every maximal ideal $p \in P$, since $A$, is a principal ideal domain. The conclusion then follows from Chapter II, § 5, no. 2, Theorem 1 (b).