Remark 1

Let $A$ be a local noetherian ring and $\Omega$ a finitely generated $A$-module. The residue field $\kappa_{\widehat{A}}$ identifies with $\kappa_A$, and the $\widehat{A}$-module $\widehat{\Omega}$ with $\widehat{A} \otimes_A \Omega$ (III, $\S$ 3, No. 4, th. 3). It then follows from A, X, p. 111, prop. 10 that the $\kappa_A$-vector space $\mathrm{Ext}_A^i(\kappa_A, \Omega)$ is canonically isomorphic to $\mathrm{Ext}_{\widehat{A}}^i(\kappa_{\widehat{A}}, \widehat{\Omega})$. Consequently, for the $A$-module $\Omega$ to be dualizing, it is necessary and sufficient that the $\widehat{A}$-module $\widehat{\Omega}$ be dualizing.