Exercise 13, § 4

Let A be a principal ideal domain with field of fractions K. a) Show that, for every A-module M, the canonical homomorphism $c_{M}: M \to D(D(M))$ is injective (cf. II, p. 389, Ex. 13). b) If $u: M \to N$ is a homomorphism of A-modules, define natural isomorphisms from $D(\operatorname{Im}(u))$ onto $\operatorname{Im}(D(u))$, from $D(\operatorname{Ker}(u))$ onto $\operatorname{Coker}(D(u))$, and from $D(\operatorname{Coker}(u))$ onto $\operatorname{Ker}(D(u))$. c) Let M be an A-module and N a submodule of M. In the notation of Sect. 9, show that $N^{00} = N$ (use a) and b)). The module $N^{0}$ has finite length if and only if $M/N$ has finite length. Conversely, for any submodule $N'$ of $D(M)$ of finite length, the module ${N'}^{0}$ is a submodule of M such that $M/N^{0}$ has finite length, and $N' = {N'}^{00}$. d) If M is a torsion A-module, and $M_{\pi}$ denotes the $\tau$-primary component of M, show that $D(M)$ is isomorphic to $\prod_{\pi \in P} \operatorname{Hom}(M_{\pi}, U_{\pi})$, where P is a system of representatives of irreducible elements of A and $U_{\pi}$ is the $\tau$-primary component of $K/A$ for each $\tau \in P$ (VII, p. 54 Ex. 3). e) Deduce from d) that $D(K/A)$ is isomorphic to the product $\prod_{\pi \in P} \hat{A}_{(\pi)}$, where $\hat{A}_{(\pi)}$ denotes the completion of the ring $A_{(\pi)}$ defined in VII, p. 54, Ex. 2, with respect to the topology defined by taking the ideals $\pi''A_{(\pi)}$ to be a basis of open neighbourhoods of 0 (this completion can also be identified with the inverse limit $\varprojlim (A/A\pi'')$).

f) If M is an A-module of finite length, and a is its annihilator, show that D(M) is isomorphic to the dual of the faithful (A/a)-module associated to M.

g) Let M be an A-module of finite length. Show that if N is a submodule of M, then there exists a submodule P of M such that M/N is isomorphic to P and M/P is isomorphic to N (use VII, p. 26, Prop. 10 and VII, p. 27, Prop. 12); two submodules of M with these properties are called reciprocal. Show that, for all a ∈ A, the submodule M(α) of M consisting of all x ∈ M such that αx = 0, and the submodule αM, are reciprocal.