See the previous post for the notion of -finitely presented modules.
Lemma. Let be a -finitely presented flat module. Then is projective.
Proof. For every prime , the module is finitely presented and flat, hence free (use Nakayama). In particular, it is projective over , hence
for all . By our previous lemma, we conclude that
for any -module , as is -finitely presented. Since is arbitrary, this forces
for any -module . Hence is projective.
Remark. Using the equational criterion for flatness, one can in fact prove that any finitely presented flat module is projective. However, I thought the above proof was nice enough to make up for this slight loss of generality.
Remark. The Stacks project gives an example of a finitely generated (but not finitely presented) flat module that is not projective.
However, there are examples of finitely generated modules over a non-Noetherian ring which are locally free and not projective. Bass settled it for non-finitely generated modules and Quillen and Suslin independently and simultaneously treated the case of finitely generated modules.