Proposition 12

Let $\rho : A \to B$ be a local homomorphism of noetherian local rings, making $B$ into a flat $A$-module. The following conditions are equivalent:

(i) $B$ is a Gorenstein ring ; (ii) $A$ and $\kappa_A \otimes_A B$ are Gorenstein rings.

Let us first treat the case where the rings $A$ and $B$ are artinian. Let $C$ denote the local ring $\kappa_A \otimes_A B$; its residue field $\kappa_C$ identifies with $\kappa_B$. Since $B$ is flat over $A$, the $B$-module $\mathrm{Hom}_B(C, B)$ is isomorphic to $\mathrm{Hom}_A(\kappa_A, A) \otimes_A B$ (I, § 2, No. 10, prop. 11), hence to $\mathrm{Hom}_A(\kappa_A, A) \otimes_{\kappa_A} C$. We deduce a sequence of isomorphisms

In particular, one has $[\mathrm{Hom}_B(\kappa_B, B) : \kappa_B] = [\mathrm{Hom}_A(\kappa_A, A) : \kappa_A] \ [\mathrm{Hom}_C(\kappa_C, C) : \kappa_C]$ and the proposition then follows from lemma 1 of No. 7.

Let us pass to the general case. If in the statement one replaces the word "Gorenstein" by the word "Macaulay", the proposition is a particular case of Prop. 10 of § 2, No. 7. We may therefore suppose that the rings $A$, $B$ and $C = \kappa_A \otimes_A B$ are Macaulay. The $B$-module $C$ is Macaulay (§ 2, No. 1, Example 4). Put $r = \dim(A)$, $s = \dim(C)$. There exists an $A$-regular sequence $(x_1, \ldots, x_r)$ of elements of $m_A$ and a sequence $(y_1, \ldots, y_s)$ of elements of $m_B$ which is regular for the $B$-module $C$ (§ 2, No. 3, Prop. 3); let $x$ denote the ideal of $A$ and $\mathfrak{y}$ the ideal of $B$ which they generate respectively. The sequence $(y_1, \ldots, y_s, \rho(x_1), \ldots, \rho(x_r))$ is $B$-regular (§ 1, No. 6, Prop. 11), and the $A$-module $B/\mathfrak{y}$ is flat (loc. cit., Prop. 10). The homomorphism of $A/x$ into $B/(xB + \mathfrak{y})$ deduced from $\rho$ by passing to quotients therefore makes $B/(xB + \mathfrak{y})$ into a flat $(A/x)$-module, and the ring $\kappa_{A/x} \otimes_{A/x} B/(xB + \mathfrak{y})$ is isomorphic to $C/\mathfrak{y}C$. The proposition thus follows from the first part of the proof, taking account of Example 3 of No. 7.