Exercise 9, § 5

Let $A$ be a discrete valuation ring, $\pi$ a uniformizer of $A$ and $K$ the field of fractions of $A$; suppose that $A$ is complete with the $\pi$-adic topology. The injective $A$-modules (Algebra, Chapter II, § 2, Exercise 11) are identical with the divisible $A$-modules (Algebra, Chapter VII, § 2, Exercise 3) and are direct sums of $A$-modules isomorphic either to $K$ or to $K/A$ (Algebra, Chapter VII, § 2, Exercise 3). Moreover, every monogenous $A$-module is isomorphic to a submodule of $K/A$. The $A$-module $\mathrm{Hom}_A(M, K/A)$ is called the (algebraic) toric dual of an $A$-module $M$ and denoted by $M^*$; the canonical mapping $c_M : M \to M^{**}$ is injective (Algebra, Chapter II, § 2, Exercise 13 (b)). For every submodule $N$ of $M$ the submodule $N^0$ of $M^*$ consisting of the $u$ such that $u(x) = 0$ is called the orthogonal of $N$ in $M^*$; the dual of $M/N$ is canonically identified with $N^0$ and the dual of $N$ with $M^*/N^0$. The toric dual of a direct limit $\varprojlim M_\alpha$ is canonically isomorphic to the inverse limit $\varprojlim M_\alpha^*$.

(a) We know that the $A$-modules of rank one (Algebra, Chapter VII, § 4,

Exercise 22) are isomorphic to a module of one of the form A, K, K/A or $A/\pi^n A$. Show that the algebraic toric duals of K and $A/\pi^n A$ are respectively isomorphic to K and $A/\pi^n A$, that the toric dual of A is isomorphic to K/A and that that of K/A is isomorphic to A (use the knowledge about the sub-A-modules of K and the fact that A is complete (with respect to the dual of K/A)). Deduce that, for every A-module M of finite rank, $M^*$ is a module of the same rank and that the canonical homomorphism $c_M$ is bijective.

(b) Let M be an A-module and N a submodule of M of finite rank. Show that the orthogonal of $N^0$ in $M^{**}$ is identified (by $c_M$) with N (use (a)).

(c) Show that an A-module M which is Noetherian (resp. Artinian) is of finite rank (embed M in its injective envelope (Algebra, Chapter II, § 2, Exercise 18)); then $M^*$ is Artinian (resp. Noetherian).

(d) Let $\sigma(M^*, M)$ denote the topology (on $M^*$) of pointwise convergence on M (K/A being given the discrete topology). Show that, if N is a submodule of M, $\sigma(N^0, M/N)$ is induced on $N^0$ by $\sigma(M^*, M)$ and that $\sigma(M^*/N^0, N)$ is the quotient by $N^0$ of the topology $\sigma(M^*, M)$. (For the second point, note that, if P is a submodule of M such that $N \subset P$, the dual of P/N is identified with $N^0/P^0$). If $M = \lim_{\longrightarrow} M_\alpha$, the topology $\sigma(M^*, M)$ is the inverse limit of the topologies $\sigma(M^*_\alpha, M_\alpha)$.

(e) The topologies $\sigma(K, K)$ and $\sigma(A, K/A)$ are the x-adic topologies; the topologies $\sigma(A/\pi^n A, A/\pi^n A)$ and $\sigma(K/A, A)$ are the discrete topologies. Deduce that for every A-module M, the module $M^*$ with $\sigma(M^*, M)$ is linearly compact (Chapter III, § 2, Exercise 15; consider M as a quotient module of a free A-module).

(f) Let M, N be two A-modules; for every homomorphism $u : M \to N$, let $^t u$ denote the homomorphism $\mathrm{Hom}(u, 1_{K/A})$ from $N^*$ to $M^*$ such that

for all $x \in M$ and all $w \in N^*$; show that $^t u$ is continuous for the topologies $\sigma(N^*, N)$ and $\sigma(M^*, M)$. If $u$ is the endomorphism $x \mapsto \pi x$ of M, $^t u$ is the endomorphism $w \mapsto \pi w$ of $M^*$. For every submodule P of M, $(u(P))^0 = ^t u(P^0)$.