Definition 1

Let $M$ be an $A$-module. The projective dimension of $M$, denoted by $\mathrm{dp}_A(M)$, is the greatest lower bound in $\overline{\mathbf{Z}}$ of the lengths of projective resolutions of $M$ (X, p. 48).

We thus have $\mathrm{dp}_A(0) = -\infty$, $\mathrm{dp}_A(M) \geq 0$ if $M \neq 0$. For $M$ to be projective, it is necessary and sufficient that $\mathrm{dp}_A(M) \leq 0$.