Exercise 11, § 3

Let A be the ring $\mathbf{Z}_p$ of p-adic integers ($p$ prime) with the p-adic topology with which it is a complete Zariski ring. Let E be the A-module $A^{(\mathbf{N})}$ with the p-adic topology. (a) Show that the completion $\hat{E}$ of E is identified with the submodule of $A^{\mathbf{N}}$ consisting of the sequences $(a_n)_{n \in \mathbf{N}}$ of elements of A such that $\lim_{n \to \infty} a_n = 0$. (b) Let $e_n$ be the n-th vector of the canonical basis of E. Consider in E the submodule F generated by the vectors $e_{2n-1}$ and the submodule G generated by the vectors $p^{2n}e_{2n} - e_{2n-1} \ (n \geq 1)$. Show that the topologies induced on F and G by the p-adic topology on E are the p-adic topologies on F and G but that, in $\hat{E}$,

(c) In $\hat{E}$, let $a_r = \sum_{n=0}^{\infty} p^n e_{r+2} \ (r \geq 0)$; let H be the submodule of $\hat{E}$ generated by the $a, (r \geq 0)$. Show that on $H$ the topology induced by the $p$-adic topology on $\hat{E}$ is the $p$-adic topology on $H$ but that $E \cap H \neq \overline{E} \cap \overline{H}$.

(d) Let $L$ be the submodule of $E$ generated by the $p^n e_n$; show that on $L$ the topology induced by the $p$-adic topology on $E$ is distinct from the $p$-adic topology on $L$.