Exercise 5, § 3

a) Let $k$ be a field, $R$ a $k$-algebra of finite degree. In order that $R$ be a Gorenstein ring, it is necessary and sufficient that the $R$-module $\mathrm{Hom}_k(R, k)$ be free of rank 1 (reduce to the case where $R$ is local, and observe that the $R$-module $\mathrm{Hom}_k(R, k)$ is injective and contains a submodule isomorphic to $\kappa_R$; for another proof, cf. § 8, No. 6, cor. of the prop. 5). b) Suppose that the above conditions are satisfied, and that in addition the algebra $R$ is graded of type $N$, with $R_0 = k$. Let $s$ be the largest integer such that $R_s \neq 0$. Prove that $R_s$ has dimension 1 over $k$, and is equal to the socle of $R$; for every nonzero $k$-linear form $\ell$ on $R_s$, the $k$-bilinear form $(x, y) \mapsto \ell(xy)$ on $R_i \times R_{s-i}$ induces for every $i$ an isomorphism of $R_i$ onto $\mathrm{Hom}_k(R_{s-i}, k)$. c) Let $A$ be a noetherian ring, $B$ an $A$-algebra which is a flat module of finite type. Prove that the following conditions are equivalent: (i) the $B$-module $\mathrm{Hom}_A(B, A)$ is projective of rank 1 ; (ii) for every maximal (resp. prime) ideal $p$ of $A$, the ring $\kappa(p) \otimes_A B$ is a Gorenstein ring.

EXERCISES

These conditions are satisfied in particular when B is a Gorenstein ring.