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

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

2. 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,

3. 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 $d = 0$,

4. 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 $d = 0$.

Proof. The equivalence of (1) and (2) is the definition of projective dimension, see Definition 10.109.2. We have (2) $\Rightarrow$ (4) by Lemma 10.109.3. The implications (4) $\Rightarrow$ (3) and (3) $\Rightarrow$ (2) are immediate. $\square$

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