The Stacks project

Proof. Assume (1). Choose a free resolution $F_\bullet \to M$ of $M$. Denote $d_ e : F_ e \to F_{e - 1}$. By Lemma 10.109.3 we see that $P_ e = \mathop{\mathrm{Ker}}(d_ e)$ is projective for $e \geq n - 1$. This implies that $F_ e \cong P_ e \oplus P_{e - 1}$ for $e \geq n$ where $d_ e$ maps the summand $P_{e - 1}$ isomorphically to $P_{e - 1}$ in $F_{e - 1}$. Hence, for any $R$-module $N$ the complex $\mathop{\mathrm{Hom}}\nolimits _ R(F_\bullet , N)$ is split exact in degrees $\geq n + 1$. Whence (2) holds. The implication (2) $\Rightarrow $ (3) is trivial.

Assume (3) holds. If $n = 0$ then $M$ is projective by Lemma 10.77.2 and we see that (1) holds. If $n > 0$ choose a free $R$-module $F$ and a surjection $F \to M$ with kernel $K$. By Lemma 10.71.7 and the vanishing of $\mathop{\mathrm{Ext}}\nolimits _ R^ i(F, N)$ for all $i > 0$ by part (1) we see that $\mathop{\mathrm{Ext}}\nolimits _ R^ n(K, N) = 0$ for all $R$-modules $N$. Hence by induction we see that $K$ has projective dimension $\leq n - 1$. Then $M$ has projective dimension $\leq n$ as any finite projective resolution of $K$ gives a projective resolution of length one more for $M$ by adding $F$ to the front. $\square$