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Is there a module which is finitely generated, torsion-free, and of projective dimension 2 at the same time? I was just wondering if such an example exists...

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In an integral domain $R$, an ideal is a torsion-free $R$-module, and if it is generated by a regular sequence of length $r$, it has projective dimension $r-1$. So take, for instance, $R=K[X,Y,Z]$ ($K$ a field) and consider the ideal $I=(X, Y, Z)$.

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  • $\begingroup$ "integral domain" is superfluous; see here. $\endgroup$ – user26857 Sep 17 '17 at 22:02
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    $\begingroup$ @user26857: I know that, but ‘Integral domain’ is for the requirement the modyule be torsion-free. I should have mentioned that. $\endgroup$ – Bernard Sep 17 '17 at 22:07

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