Definition 1

Let $A$ be a noetherian ring. One says that a $A$-module $\Omega$ is dualizing if it is finitely generated and if, for every maximal ideal $m$ of $A$, the $A/m$-vector space $\mathrm{Ext}_A^i(A/m, \Omega)$ is zero for $i \neq \mathrm{ht}(m)$ and of dimension 1 for $i = \mathrm{ht}(m)$.

For every maximal ideal $m$ of $A$ and every integer $i$, the $A/m$-vector space $\mathrm{Ext}_A^i(A/m, \Omega)$ is canonically isomorphic to $\mathrm{Ext}_{A_m}^i(A/m, \Omega_m)$ ($\S$ 3, No. 2, prop. 2). Consequently, for a finitely generated $A$-module $\Omega$ to be dualizing, it is necessary and sufficient that the $A_m$-module $\Omega_m$ be dualizing for every maximal ideal $m$ of $A$.