Exercise 4, § 9

Let $A$ be a local Macaulay ring of dimension $d$; let $\Omega$ be a finitely generated $A$-module, of finite injective dimension, of depth $d$, such that the canonical homomorphism $A \to \mathrm{End}_A(\Omega)$ is bijective. Prove that the $A$-module $\Omega$ is dualizing (reason by induction on $d$, deducing from Exercise 7 of § 3 that for every $\Omega$-regular element $x$ of $m_A$, the canonical homomorphism $A/xA \to \mathrm{End}_{A/xA}(\Omega/x\Omega)$ is bijective).