Exercise 12, § 9

Let $k$ be a field, $R$ a graded $k$-algebra of type $N$, such that $R_0 = k$ and that $R$ is a Macaulay ring, of dimension $d$. For every graded $R$-module of finite type $M$, denote by $P_M(T)$ the Poincaré series $\sum_i (\dim_k M_i) T^i$, and put

it is an element of $\mathbf{Z}[T, T^{-1}]$ (VIII, § 6, No. 3, Prop. 5).

(a) Let $\Omega$ be a dualizing $R$-module; prove that $\Omega$ admits a structure of graded $R$-module for which $Q_{\Omega}(T) = (-1)^d Q_R(T^{-1})$ (write $R$ as a quotient of a polynomial algebra $A = k[X_1, \ldots, X_n]$, with $\deg(X_i) = d_i$; if $L$ is a graded free resolution of the $A$-module $R$, observe that the complex $\mathrm{Hom}_{\mathbf{A}}(L, A)(-d)$ defines one of $\Omega$, and use Example 3 of VIII, § 4, No. 2).

(b) Suppose that $R$ is a Gorenstein ring; prove that there exists an integer $a \in \mathbf{Z}$ satisfying $Q_R(T^{-1}) = (-1)^d T^a Q_R(T)$.

(c) Suppose that $R$ is integral and that there exists an integer $a \in \mathbf{Z}$ such that one has $Q_R(T^{-1}) = (-1)^d T^a Q_R(T)$. Prove that $R$ is a Gorenstein ring (endow $\Omega$ with a grading for which $Q_{\Omega} = Q_R$, and prove that a nonzero element of $\Omega_0$ forms a basis of $\Omega$).

(d) Let $R$ be the graded $k$-algebra $k[X, Y]/(X^3, XY, Y^2)$; prove that one has $Q_R(T^{-1}) = T^{-2} Q_R(T)$, but that $R$ is not a Gorenstein ring.