Exercise 1, § 7

a) Show that the ring $\mathbf{Z}_p$ of $p$-adic integers [§ 6, Exercise 23 a)] is isomorphic to the inverse limit of the sequence of discrete rings $\mathbf{Z}/(p^n)$, $f_{nm}: \mathbf{Z}/(p^m) \to \mathbf{Z}/(p^n)$ for $n \leq m$ being the canonical homomorphism when $\mathbf{Z}/(p^n)$ is considered as a quotient ring of $\mathbf{Z}/(p^m)$.

b) For each integer $n > 0$, let $G_n$ be the group $\mathbf{Z}$ of rational integers, and let $X_n$ be its quotient group $\mathbf{Z}/(p^n)$ ($p$ prime), $G_n$ and $X_n$ carrying the discrete topology; consider $(G_n)$ as an inverse system, $G_m \to G_n$ being the identity mapping; and consider $(X_n)$ as an inverse system, $f_{nm}: X_m \to X_n$ being the canonical homomorphism (see a) for $n \leq m$. Then the orbit of each point of $X_n$ with respect to $G_n$ is compact, but the orbit of a point $x = (x_n)$ of $X = \varprojlim X_n$ with respect to $G = \varprojlim G_n$ is not compact, and is not isomorphic to the inverse limit of the orbits of the $x_n$; also the canonical mapping $X/G \to \varprojlim X_n/G_n$ is not injective.

c) For each $n > 0$, let $G_n$ be the subgroup $p^n\mathbf{Z}$ of $\mathbf{Z}$ ($p$ prime), and let $X_n$ be the group $\mathbf{Z}$, where $G_n$ and $X_n$ carry the discrete topology and $G_n$ operates by translations on $X_n$. Consider $(G_n)$ and $(X_n)$ as inverse systems, $X_m \to X_n$ being the identity mapping and $G_m \to G_n$ the canonical injection $(n \leq m)$. Then the stabilizer of each point of $X_n$ (with respect to $G_n$) is compact, the orbit of each point of $X = \varprojlim X_n$ with respect to $G = \varprojlim G_n$ is compact, but the canonical mapping $X/G = \varprojlim X_n/G_n$ is not surjective.

d) Construct, using b) and c), an example of an inverse system of spaces with operators $(X_\alpha)$ with respect to an inverse system of groups $(G_\alpha)$, such that, if $X = \varprojlim X_\alpha$ and $G = \varprojlim G_\alpha$, the canonical mapping $X/G \to \varprojlim X_\alpha / G_\alpha$ is neither injective nor surjective.