Lemma 1

Let $A$ be an artinian local ring. The following conditions are equivalent:

(i) $A$ is a Gorenstein ring; (ii) the $A$-module $A$ is injective; (iii) the vector space $\mathrm{Hom}_A(\kappa_A, A)$ over $\kappa_A$ is of dimension 1.

Let us recall (A, VIII, p. 3.5) that $A$ has a minimal nonzero ideal; such an ideal is a simple module, hence isomorphic to $\kappa_A$. The vector space $\mathrm{Hom}_A(\kappa_A, A)$ over $\kappa_A$ is therefore nonzero; saying that it is of dimension 1 means that $A$ contains only one minimal nonzero ideal, which is therefore the socle of $A$ (A, VIII, § 4, No. 6).

The equivalence of (i) and (ii) was proved after Prop. 10. Suppose that the $A$-module $A$ is injective. Let $x$ and $y$ be two nonzero elements of $A$ annihilated by $m_A$. There exists a unique $A$-linear mapping $\varphi : Ax \to A$ such that $\varphi(x) = y$; since $A$ is injective, it extends to an endomorphism of $A$, which implies that $y$ belongs to $Ax$, whence (iii).

Conversely, suppose that $\mathrm{Hom}_A(\kappa_A, A)$ is of dimension 1. Let $M$ be a finitely generated $A$-module; it is of finite length, hence has a composition series whose quotients are isomorphic to $\kappa_A$. One deduces by induction on the length of $M$ the inequality $\mathrm{long}_A(\mathrm{Hom}_A(M, A)) \leqslant \mathrm{long}_A(M)$. In the exact sequence of extension modules

one therefore has $\mathrm{long}_A(\mathrm{Hom}_A(m_A, A)) \leqslant \mathrm{long}_A(m_A) = \mathrm{long}(A) - 1 = \mathrm{long}_A(\mathrm{Im}\, \alpha)$. Consequently $\alpha$ is surjective, $\mathrm{Ext}_A^1(\kappa_A, A)$ is zero and the $A$-module $A$ is injective (No. 3, Prop. 4).