Exercise 10, § 5

The hypotheses and notation are the same as in Exercise 9.

(a) If M is a discrete topological A-module, show that M is a torsion module. If further M is linearly compact, show that M is Artinian (if N is the kernel of the endomorphism $x \mapsto \pi x$, observe that N is linearly compact and discrete and can be considered as a vector $(A/\pi A)$-space; then use Exercise 20 (d) of Chapter II, § 2).

(b) Deduce from (a) that every linearly compact topological A-module is strictly linearly compact (Chapter II, § 2, Exercise 19).

(c) The topological toric dual of a topological A-module M is the submodule $M'$ of the algebraic toric dual $M^*$ consisting of the continuous homomorphisms from M to K/A (the latter being given the discrete topology). If the topology on M is linear (Chapter II, § 2, Exercise 14) and N is a closed submodule of M, show that the topological toric dual of M/N is identified with $N^0 \cap M'$ and that of N with $M'/(N^0 \cap M')$ (to determine the dual of N, note that, for every continuous homomorphism $u$ from N to K/A, there exists an open submodule U of M such that $u(x) = 0$ in $N \cap U$, then use Exercise 9).

(d) For a topological A-module M of finite rank, the topological toric dual M' is equal to the algebraic toric dual M* (reduce it the case of modules of rank one).

(e) Let M be a Hausdorff topological A-module whose topology is linear; show that for all $x \neq 0$ in M, there exists $u \in M'$ such that $u(x) \neq 0$ (note that there exists an open submodule U of M such that $x \notin U$); in other words, the canonical mapping $M \to (M')^*$ is injective. Deduce that, if N is a closed submodule of M, $N^0 \cap M'$ is dense in $N^0$ with the topology $\sigma(M^*, M)$ (use (d)), and consequently $N = M \cap (N^0 \cap M')^0$ in $M^{**}$.

Show that M is dense in $(M')^*$ with the topology $\sigma((M')^*, M')$.

(f) Let M be a linearly compact topological A-module; show that the canonical injection $M \to (M')^*$ is a bijection and that the topology on M (identified with $(M')^*$) is equal to $\sigma(M, M')$. (Observe that, if U is an open submodule of M, M/U is Artinian by (a) and therefore $U^0$ is Noetherian (Exercise 9 (c)) and deduce the identities of the topologies considered on M; complete with the aid of (e).)

(g) Let M, N be two topological A-modules whose topologies are linear; for every continuous homomorphism $u : M \to N$, $^t u(N') \subset M'$ and $^t u$ is also used (by an abuse of notation) to denote the linear mapping from N' to M' with the same graph as $^t u$. If M and N are Hausdorff, show that the restriction to M of $^t (t u)$ coincides with $u$ (M and N being respectively considered as canonically embedded in $(M')^*$ and $(N')$; moreover, for every submodule Q of N,

(use (e)).

(h) Let M be a linearly compact A-module; deduce from (g) and Exercise 9 (f) that for M to be torsion-free, it is necessary and sufficient that M' be divisible and that for M to be divisible, it is necessary and sufficient that M' be torsion-free.