Exercise 1, § 9

Let $A$ be a local Macaulay ring. a) Suppose that $A$ admits a dualizing module $\Omega$; equip the $A$-module $A \oplus \Omega$ with the structure of $A$-algebra for which $\Omega$ is a square zero ideal. Prove that $A \oplus \Omega$ is a Gorenstein ring (using a maximal sécante sequence, reduce to the case where $A$ is artinian; observe that the socle of $A \oplus \Omega$ identifies with that of the $A$-module $\Omega$). b) Deduce that for $A$ to admit a dualizing module, it is necessary and sufficient that there exist a Gorenstein ring $B$ and a surjective homomorphism from $B$ onto $A$.