Exercise 9, § 8

Soit $A$ a local noetherian ring satisfying the condition (GM) ($§ 3$, Exercise 13). a) Let $I$ be a complex of injective $A$-modules, of length $\ell$, whose homology is non-zero and of finite type. Prove the inequality $\ell \geq \dim(A)$ (reduce to loc. cit. a) by Matlis duality). b) If there exists an $A$-module $M$ of finite type and of finite injective dimension, $A$ is a Macaulay ring (apply $a$) to an injective resolution of $M$). c) In order that $A$ be a Gorenstein ring, it is necessary and sufficient that it possess a monogenic module of finite injective dimension (use Exercise 8, $b$).