An exercise in "Commutative algebra with a view towards Algebraic geometry" by Eisenbud states that a torsion-free module over a Dedekind domain is a projective module (see page $484$, Exercise $19.6$). But I am not able to prove this or find a reference for the result (there are several references for the case when the module is also finitely generated). Can somebody give a hint or a reference for the result?
2026-03-26 12:53:51.1774529631
Are torsion-free modules over principal ideal domains/Dedekind domains projective
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This isn't true at all. For instance, $\mathbb{Q}$ is torsion-free but not projective over $\mathbb{Z}$. I would assume that there's just a typo in Eisenbud and that exercise means to assume the module is finitely generated.