Let $A$ be a commutative ring with unity and $M$ an $A$-module. Show that $M$ is flat if and only if $M_\mathfrak{m}$ is a flat $A_\mathfrak{m}$-module for all maximal $\mathfrak{m}\subseteq A$. If, in addition, $A$ is a Dedekind domain, show that a finitely generated torsion-free $A$-module is projective. Furthermore, show that any finitely generated $A$-module $M$ is the direct sum of a projective $A$-module and a sum of cyclic torsion modules.
How is this problem done? What is the relationship between the three parts?
The first part follows from the usual argument with $\text{Tor}$. It's clear that if $M$ is flat, then so is $M_{\mathfrak{m}}$ for all maximal ideals $\mathfrak{m}$. Conversely, if $M_\mathfrak{m}$ is flat for all maximal ideals $\mathfrak{m}$, then for any $A$-module $N$
$$\text{Tor}_1^R(M,N)_\mathfrak{m}=\text{Tor}_1^{R_\mathfrak{m}}(M_\mathfrak{m},N_\mathfrak{m})=0$$
so, since being zero is a local property, $\text{Tor}_1^R(M,N)=0$. Since this was true for all $N$, $M$ is flat.
If $M$ is finitely generated and torsion free, then $M_\mathfrak{m}$ is a finitely generated torsion free $A_\mathfrak{m}$-module. Since $A$ is a Dedekind domain, this implies that $A_\mathfrak{m}$ is a DVR, and so $M_\mathfrak{m}$ is free (and in particular flat) by the structure theorem.
I'd like to hear your ideas for the last part, before I chip in.