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Let $G$ be a finite group, $\mathbb{Z}_{(p)}$ be the ring of p-local integers (localization of $\mathbb{Z}$ at $p\mathbb{Z}$). Let $M$ be a $p$-torsion free (i.e. $pm = 0$ implies $m=0$) $\mathbb{Z}_{(p)}G$-module with finite projective dimension. Does it follow that $M$ must be projective itself?

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