Corollary 3

Let M be a finitely generated A-module, and let i be an integer. The A-module $\operatorname{Ext}_A^i(M, \Omega)$ is finitely generated, and its support is of codimension $\geqslant i$ in Spec(A).

The first assertion follows from A, X, p. 108, corollary. Let $\mathfrak{p}$ be a prime ideal of the support of $\operatorname{Ext}_A^i(M, \Omega)$. One has $\operatorname{Ext}_A^i(M, \Omega)_\mathfrak{p} \neq 0$, hence $\operatorname{Ext}_{A_\mathfrak{p}}^i(M_\mathfrak{p}, \Omega_\mathfrak{p}) \neq 0$ (§ 3, No. 2, proposition 2), which implies $\operatorname{di}_{A_\mathfrak{p}}(\Omega_\mathfrak{p}) \geqslant i$. Since $\Omega_\mathfrak{p}$ is a dualizing $A_\mathfrak{p}$-module (proposition 2), one has $\operatorname{di}_{A_\mathfrak{p}}(\Omega_\mathfrak{p}) = \dim(A_\mathfrak{p})$ (proposition 1), whence the corollary.