Corollary 2

Suppose that the ring $A$ is presentable. The set of prime ideals $p$ of $A$ such that $A_p$ is a Gorenstein ring is open in $\mathrm{Spec}(A)$.

Let $p$ be a prime ideal of $A$ such that $A_p$ is a Gorenstein ring. Then it is a Macaulay ring; replacing $A$ if necessary by $A_f$, where $f$ is a suitable element of $A - p$, one is reduced to the case where $A$ is a Macaulay ring ($§ 4$, No. 4, Proposition 7, c)). Let $\Omega$ be a dualizing $A$-module (No. 3, Corollary 3 of Proposition 6). Then $\Omega_p$ is a dualizing module over the Gorenstein ring $A_p$ (No. 1, Proposition 2), hence is free of rank 1 (Corollary 1). It follows that there exists an element $g$ of $A - p$ such that the $A_g$-module $\Omega_g$ is free of rank 1 (II, $§ 5$, No. 1, Corollary of Proposition 2). Thus $A_g$ is a Gorenstein ring (Corollary 1) and the same is true of $A_q$ for every prime ideal $q$ of $A$ not containing $g$ ($§ 3$, No. 7, Example 1), which proves the corollary.