Finite torsion-free modules over Dedekind domains are projective

Choose a surjection $\pi :F\to M$ from a finite free module $F$. For a maximal ideal $\mathfrak{m}$, the localisation $R_{\mathfrak{m}}$ is a DVR by [L1], hence a PID, and $M_{\mathfrak{m}}$ is still finitely generated and torsion-free by [L3]. Therefore [L2] makes $M_{\mathfrak{m}}$ a free $R_{\mathfrak{m}}$-module. Thus the localised surjection ${\pi }_{\mathfrak{m}}:F_{\mathfrak{m}}\to M_{\mathfrak{m}}$ splits, and clearing the finitely many denominators in one local section yields $t_{\mathfrak{m}}\mathrm{\notin }\mathfrak{m}$ and a global map $u_{\mathfrak{m}}:M\to F$ such that $\pi u_{\mathfrak{m}}=t_{\mathfrak{m}}{\mathrm{id}}_M$.

Let $J:=\{r\in R:\text{there exists }u:M\to F\text{ with }\pi u=r{\mathrm{id}}_M\}$. This is an ideal of $R$, and step 1.1 shows that for every maximal ideal $\mathfrak{m}$ one has $t_{\mathfrak{m}}\in J\setminus \mathfrak{m}$. Therefore $J$ is not contained in any maximal ideal. By [L5], $J$ cannot be proper, so $1\in J$. Choose $u:M\to F$ with $\pi u={\mathrm{id}}_M$. Then $\pi$ splits, and [L4] makes $M$ projective.