Exercise 2, § 8

Let $A$ be a noetherian local complete ring, $M$ an $A$-module. Now prove that the following conditions are equivalent:

(i) the canonical homomorphism $\alpha_M : M \to D(D(M))$ is bijective ; (ii) there exists an exact sequence $0 \to M' \to M \to M'' \to 0$, where $M'$ is a finitely generated module and $M''$ artinian.

(If $M$ satisfies (i), show that its socle is of finite dimension over $\kappa_A$, and that the same is true of every quotient of $M$ ; construct a finitely generated submodule $N$ of $M$ such that $M/mN$ is essential extension of its socle, and use Exerc. 1).