It's been a while I get stuck on proof from Lang's Algebra book, which states that the completion and the inverse limit $\varprojlim G/H_{r}$ are isomorphic where $\{H_r \}$ is a sequence of normal subgroups in $G$ with $H_r \supset H_{r+1}$ for all $r$.
Theorem 10.1: The completion and the inverse limit $\varprojlim G/H_{r}$ are isomorphic under natural mappings.
Proof: We give the maps. Let $x=\{x_n\}$ be a Cauchy sequence. Given $r$, for all $n$ sufficiently large, by the definition of Cauchy sequence, the class of $x_n \mod H_r$ is independent of $n$. Let this class be $x(r)$. Then the sequence $(x(1),x(2),…)$ defines an element of the inverse limit. Conversely, given an element $(\overline{x}_1,\overline{x}_2,…)$ in the inverse limit, with $\overline{x}_n \in G/H_n$, let $x_n$ be a representative in G. Then the sequence $\{x_n\}$ is Cauchy. We leave to the reader to verify that the maps we have defined are inverse isomorphisms between the completion and the inverse limit.
Actually, I don't understand this proof at all. Given $r$ and $x=\{x_n\}$ be a Cauchy sequence, what is exactly $x(r)$? By definition of Cauchy sequence, there is exits $N$ such that for all $m,n \geqslant N $ we have $x_nx_m^{-1} \in H_r$, so is $x(r)$ the set $\{ x_i \mid i \geqslant N\}$?
In the proof, he said that the sequence $(x(1),x(2),…)$ defines an element of the inverse limit. Why does $(x(1),x(2),…) $ modulo the null sequences give us a single element?
To see that $\{x_n \}$ is a Cauchy sequence from $(\overline{x}_1,\overline{x}_2,…)$, by definition of limitt inverse, we have $f^{n}_{m}(\overline{x}_n)=\overline{x}_m$, i.e $x_m, x_n$ are equal in $G/H_m$ which means $x_m x_n^{-1} \in H_m$. Therefore, given $r$, for any $m,n \geqslant r$ we have $x_m x_n^{-1} \in H_m \subset H_r$ so $\{x_n \}$ is indeed a Cauchy sequence. But how does two representative sequence $\{x_n \}$ and $\{x'_n \}$ of $(\overline{x}_1,\overline{x}_2,…)$ give us a single element in completion?
Any helps would be appreciated! Thanks.
Best Answer
If you have a Cauchy sequence $x = \{x_n\}$, then for each $r$, the sequence $\{x_n H_r\}$ becomes eventually constant and so we can set $x(r) = x_n H_r$ where $n$ is such that $x_n H_r = x_m H_r$ for all $m \geq n$. This clearly defines an element $(x(1),x(2),\dots)$ in $\varprojlim G/H_{r}$.
Now, if $y = \{y_n\}$ is another Cauchy sequence differing from $\{x_n\}$ by a null sequence, then for each $r$, you have $x_n H_r = y_n H_r$ for large $n$, so the elements $x(r)$ and $y(r)$ in $G/H_r$ coincide.
For the other issue, given two representative sequences $\{x_n\}$ and $\{x_n'\}$ you have $x_n H_n = x_n' H_n$ for all $n$. It follows that $x_n^{-1}x_n' \in H_r$ for all $n \geq r$ and thus $\{x_n\}$ and $\{x_n'\}$ differ by a null sequence.