Remark

Let us retain the assumptions of cor. 1 and suppose further that the canonical homomorphism $\rho : A \to B$ is injective. Then one has $\dim(A_m) = \dim(B_m)$ for every maximal ideal $m$ of $A$ (VIII, $§ 2$, No. 3, th. 1 a)). By prop. 6 and cor. 1, $\mathrm{Ext}_A^i(B, \Omega)$ is zero for $i \neq 0$, and the $B$-module $\mathrm{Hom}_A(B, \Omega)$ is dualizing.