Lemma 10.109.6. Let $R$ be a Noetherian ring. Let $M$ be a finite $R$-module. Let $d \geq 0$. The equivalent conditions (1) – (4) of Lemma 10.109.4 are also equivalent to

there exists a resolution $0 \to P_ d \to P_{d - 1} \to \ldots \to P_0 \to M \to 0$ with $P_ i$ finite projective.

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