Lemma 10.109.4. Let $R$ be a ring. Let $M$ be an $R$-module. Let $d \geq 0$. The following are equivalent

$M$ has projective dimension $\leq d$,

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

for some resolution $\ldots \to P_2 \to P_1 \to P_0 \to M \to 0$ with $P_ i$ projective we have $\mathop{\mathrm{Ker}}(P_{d - 1} \to P_{d - 2})$ is projective if $d \geq 2$, or $\mathop{\mathrm{Ker}}(P_0 \to M)$ is projective if $d = 1$, or $M$ is projective if $d = 0$,

for any resolution $\ldots \to P_2 \to P_1 \to P_0 \to M \to 0$ with $P_ i$ projective we have $\mathop{\mathrm{Ker}}(P_{d - 1} \to P_{d - 2})$ is projective if $d \geq 2$, or $\mathop{\mathrm{Ker}}(P_0 \to M)$ is projective if $d = 1$, or $M$ is projective if $d = 0$.

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