Lemma 10.109.9. Let $R$ be a ring. Let $0 \to M' \to M \to M'' \to 0$ be a short exact sequence of $R$-modules.

1. If $M$ has projective dimension $\leq n$ and $M''$ has projective dimension $\leq n + 1$, then $M'$ has projective dimension $\leq n$.

2. If $M'$ and $M''$ have projective dimension $\leq n$ then $M$ has projective dimension $\leq n$.

3. If $M'$ has projective dimension $\leq n$ and $M$ has projective dimension $\leq n + 1$ then $M''$ has projective dimension $\leq n + 1$.

Proof. Combine the characterization of projective dimension in Lemma 10.109.8 with the long exact sequence of ext groups in Lemma 10.71.7. $\square$

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