Lemma 10.77.3. Let $R$ be a Noetherian ring. Let $P$ be a finite $R$-module. If $\mathop{\mathrm{Ext}}\nolimits ^1_ R(P, M) = 0$ for every finite $R$-module $M$, then $P$ is projective.

Proof. Choose a surjection $R^{\oplus n} \to P$ with kernel $M$. Since $\mathop{\mathrm{Ext}}\nolimits ^1_ R(P, M) = 0$ this surjection is split and we conclude by Lemma 10.77.2. $\square$

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