Corollary 2

Let $A$ be a noetherian ring, $J$ an ideal of $A$, $\hat{A}$ the separated completion of $A$ for the $J$-adic topology. Consider the following conditions:

(i) $A$ is a Gorenstein ring; (ii) $\hat{A}$ is a Gorenstein ring; (iii) for every maximal ideal $m$ of $A$ containing $J$, $A_m$ is a Gorenstein ring; (iv) for every prime ideal $p$ of $A$ such that $p + J \neq A$, $A_p$ and $\kappa(p) \otimes_A \hat{A}$ are Gorenstein rings.

Conditions (ii) to (iv) are equivalent, and are implied by (i). When the ideal $J$ is contained in the radical of $A$, conditions (i) to (iv) are equivalent.

This corollary is deduced from cor. 1 in the same way as corollary 4 of prop. 9 of § 2, No. 7, was deduced: it is enough in the proof to replace the word "Macaulay" by the word "Gorenstein".