Exercise 2, § 9

Let $A$ be a regular local ring, $I$ an ideal of $A$, $B$ the ring $A/I$; suppose that $B$ is a Macaulay ring. Let $(L, p)$ be a minimal projective resolution of the $A$-module $B$; it is of length $c = \mathrm{ht}(I)$. (a) Let $L^*$ be the complex $\mathrm{Homgr}_A(L, A)$; prove that one has $H^i(L^*) = 0$ for $i \neq c$ and that the $B$-module $H^c(L^*)$ is dualizing. (b) Prove that the following conditions are equivalent: (i) $A$ is a Gorenstein ring ; (ii) the complexes $L$ and $L^*(-d)$ are isomorphic ; (iii) $L_c$ is free of rank one.