Proposition 11

Let $A$ be a noetherian ring. The following conditions are equivalent:

(i) $A$ is a Gorenstein ring ; (ii) for every prime ideal $p$ of $A$, the $\kappa(p)$-vector space $\operatorname{Ext}^i_{A_p}(\kappa(p), A_p)$ is zero for $i \neq \operatorname{ht}(p)$ and of dimension 1 for $i = \operatorname{ht}(p)$. (iii) for every maximal ideal $m$ of $A$, there exists an integer $i > \operatorname{ht}(m)$ such that $\operatorname{Ext}^i_{A_m}(A/m, A_m) = 0$.

(i) $\Rightarrow$ (ii) : this follows from th. 2 applied to the local Gorenstein ring $A_p$ (example 1). (ii) $\Rightarrow$ (iii) : this is trivial. (iii) $\Rightarrow$ (i) : under assumption (iii), $A_m$ is a Gorenstein ring for every maximal ideal $m$ of $A$ (th. 2), and $A$ is Gorenstein.