Exercise 13, § 3

Let M be an A-module. A minimal injective resolution of M is an injective resolution (e, l) of M such that l' is an injective envelope of $Z^n(l)$ for all $n \geqslant 0$. (a) Every A-module admits minimal injective resolutions; two such resolutions are isomorphic. (b) Let l, l' be two injective resolutions of M, l being minimal; let $f : l \to l'$ and $g : l' \to l$ be two morphisms of resolutions. Then f is injective, g is surjective and l' is the direct sum of subcomplexes lm (f) (isomorphic to l) and Ker (g) (with zero homology).