Remark 3

Let $A$ be a noetherian ring, $M$ a finitely generated $A$-module of finite projective dimension, $N$ a finitely generated $A$-module whose support is equal to $\mathrm{Spec}(A)$. According to the preceding remark and Props. 2 and 3 of No. 2, the projective dimension $n$ of $M$ is the largest integer $i$ such that $\mathrm{Ext}_A^i(M, N)$ is nonzero; the support of the $A$-module $\mathrm{Ext}_A^n(M, N)$ is the set of elements $p$ of $\mathrm{Spec}(A)$ such that $\mathrm{dp}_{A_p}(M_p) = n$.

There may exist finitely generated $A$-modules $M$ of projective dimension $+\infty$, satisfying $\mathrm{Ext}_A^i(M, A) = 0$ for $i$ sufficiently large: this is the case for example of the $A$-module $\kappa_A$ when $A$ is a Gorenstein local ring which is not regular.