Exercise 3, § 4

Let $A$ be an integrally closed Noetherian domain, $K$ its field of fractions, $M$ a reflexive lattice of a vector space of finite rank over $K$ and $L$ a free lattice of $V$ containing $M$. Show that there exists a free lattice $L_0$ of $V$ such that $M = L \cap L_0$. (Consider the finite set $I$ of prime ideals $\mathfrak{p}$ of $A$ of height 1 such that $L_0 \neq M_\mathfrak{p}$ and the principal ideal domain $S^{-1}A$, where $S = \bigcap_{\mathfrak{p} \in I} (A - \mathfrak{p})$; show that there exist a free lattice $L_0$ of $V$ such that $M_0 = (L_0)_\mathfrak{p}$ for all $\mathfrak{p} \in I$ and an $s \in S$ such that $M \subset s^{-1}L_0 \subset L_1$.)