Corollary 1

Let A be a noetherian ring, $\Omega$ a dualizing A-module, and B a finite A-algebra; suppose that the A-module B is Macaulay. The B-module $\mathrm{Ext}_A(B, \Omega)$ is dualizing.

Let $\Omega'$ denote the $B$-module $\mathrm{Ext}_A(B, \Omega)$. Let $n$ be a maximal ideal of $B$; its inverse image in $A$ is a maximal ideal $m$ (V, $§ 2$, No. 1, prop. 1). The $A_m$-algebra $B_m = A_m \otimes_A B$ is finite, and is a Macaulay $A_m$-module; by the proposition, the $B_m$-module $\Omega'_m$, which identifies with $\mathrm{Ext}_{A_m}(B_m, \Omega_m)$ ($§ 3$, No. 2, prop. 2), is dualizing. Since $B_n$ is a ring of fractions of $B_m$, the $B_n$-module $\Omega'_n$ is dualizing, whence the corollary.