Exercise 5, § 8

Let $A$ be a noetherian local ring, $I$ a Matlis $A$-module. For every $A$-module $M$ we denote by $D(M)$ the Matlis dual $\mathrm{Hom}_A(M, I)$. We recall that $\mathrm{dpl}_A(M)$ denotes the greatest lower bound of the lengths of the flat resolutions of $M$ (A, X, p. 202, exer. 7). Prove the equalities $\mathrm{di}_A(D(M)) = \mathrm{dpl}_A(M)$ , $\mathrm{di}_A(M) = \mathrm{dpl}_A(D(M))$.