Exercise 4, § 8

Let $A$ be a noetherian ring, $M$ a finitely generated $A$-module. For all $\mathfrak{p} \in \mathrm{Spec}(A)$, we put $\mu^i(\mathfrak{p}, M) = \dim_{k(\mathfrak{p})} \mathrm{Ext}^i_{A_{\mathfrak{p}}}(k(\mathfrak{p}), M_{\mathfrak{p}})$ ; we denote by $(I(\mathfrak{p}), e_{\mathfrak{p}})$ an injective envelope of the $A$-module $A/\mathfrak{p}$. Let $I$ be a minimal injective resolution of $M$ (which means that $I^n$ is an injective envelope of $Z^n(I)$ for all $n \geqslant 0$, cf. A, X, p. 182, Exercise 13). Prove that the $A$-module $I^p$ is isomorphic to $\bigoplus_{\mathfrak{p}} I(\mathfrak{p})^{\mu^p(\mathfrak{p}, M)}$.