Remark 2

Let A be a noetherian local ring, M a finitely generated A-module of finite projective dimension $n$, N a nonzero finitely generated A-module. The A-module $\operatorname{Ext}_A^n(M, N)$ is nonzero: let in fact L be a minimal projective resolution of M, and d its differential. We have an exact sequence

Since $d_n \otimes 1_{\kappa_A}$ is zero, one deduces from this, by tensor product with $\kappa_A$, an isomorphism $\kappa_A \otimes_A \mathrm{Hom}_A(L_n, N) \longrightarrow \kappa_A \otimes_A \mathrm{Ext}_A^n(M, N)$, whence, taking account of formula (1) above,

which is nonzero by Prop. 4 and Nakayama's lemma. Consequently, the projective dimension of M is the largest integer $i$ such that $\mathrm{Ext}_A^i(M, N)$ is nonzero.