Exercise 15, § 3

Let $A$ be a local ring noetherian, $M$ a $A$-module finitely generated and of finite injective dimension.

a) Let $p \in \operatorname{Supp}(M)$. Prove the equality $\operatorname{prof}(A) = \operatorname{prof}(A_p) + \dim(A/p)$. (Put $p = \operatorname{prof}(A_p)$ and $q = \dim(A/p)$; observe that one has $p = \operatorname{di}_{A_p}(M_p)$, whence $\operatorname{Ext}^{p}_{A_p}(\kappa(p), M_p) \neq 0$ and consequently $\operatorname{Ext}^{p+q}_{A}(\kappa_A, M) \neq 0$ by Lemma 3 of § 1, No. 7, which entails $\operatorname{prof}(A) = \operatorname{di}_A(M) = p + q .$) If $\operatorname{Supp}(M) = \operatorname{Spec}(A)$, $A$ is a Macaulay ring.

b) Prove the equality $\operatorname{grade}(M) + \dim(M) = \operatorname{prof}(A)$ (use a) and Exercise 15 of § 1).