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In his answer to this question, Andrea claims that if $A \subset B$ is an extension of rings of integers of number fields, $B$ is locally free over $A$.

How can one prove this?

Furthermore, I am looking for an example (with $A$ and $B$ as above) where $B$ is not a free $A$-module (in case $A = \mathbb{Z}$, $B$ is always free over $A$, since it is a finitely generated, torsion-free $\mathbb{Z}$-module).

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The extension $A\subseteq B$ is always finite (because $B$ is finite over $\mathbf{Z}$). Since $B$ is a torsion-free $A$-module, it is $A$-flat (since $A$ is Dedekind). Over a Noetherian ring, finite flat is the same as finite locally free.

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    although the $n$otio$n$s coincide for finitely presented modules.2012-09-04