1. ## Inverse Limit

Let $(a_n)_{n\in \Bbb{N}}, (b_n)_{n\in \Bbb{N}}$ be two sequences of positive integers such that for all $n \in \Bbb{N}$, $a_n$ divides (but is not equal to) $a_{n+1}$ and $b_n$ divides but is not equal to $b_{n+1}$. Let $X_n = \prod_{i=1}^n \Bbb{Z}/a_n\Bbb{Z}$. Define maps $X_m \mapsto X_n$ for all $m >n$ such that $(x_1,\ldots x_m) \mapsto (x_{m-n}, \ldots, x_m)$ (by the usual mappings). Also, define maps $\phi_n: X_n \to \Bbb{Z}/b_n\Bbb{Z}$ such that the following diagram commutes:

$\begin{matrix} X_m & \stackrel{\phi_m}{\longrightarrow} & \Bbb{Z}/b_m\Bbb{Z} \\ \downarrow & & \downarrow \\ X_n & \stackrel{\phi_n}{\longrightarrow} & \Bbb{Z}/b_n\Bbb{Z}\end{matrix}$

Where the down arrows are the maps I defined above for the $X_m \mapsto X_n$ and the standard maps for $\Bbb{Z}/b_m\Bbb{Z} \mapsto \Bbb{Z}/b_n\Bbb{Z}$.

Let $X = \varprojlim X_n$ and $B=\varprojlim \Bbb{Z}/b_n\Bbb{Z}$. Does there exist a map $\phi: X \to B$ such that the following diagram commutes:

$\begin{matrix} X & \stackrel{\phi}{\longrightarrow} & B \\ \downarrow & & \downarrow \\ X_n & \stackrel{\phi_n}{\longrightarrow} & \Bbb{Z}/b_n\Bbb{Z}\end{matrix}$

2. ## Re: Inverse Limit

I guess I should ask does the map $X \to X_n$ even make sense? I mean, $X$ is essentially an infinite-tuple, and the map would require taking the last $n$ positions... but that might not even be possible. Perhaps the inverse limit of $X_n$ doesn't even exist...

3. ## Re: Inverse Limit

An easier example:

Define a map from $\Bbb{Z}^m \to \Bbb{Z}^n$ for all $m > n$ by $(x_1,\ldots,x_m) \mapsto (x_{m-n+1},\ldots, x_m)$. Does $\varprojlim \Bbb{Z}^n$ exist given those maps?

4. ## Re: Inverse Limit

Oh, and for my first example, I have

$0 \to (\varprojlim \Bbb{Z}/b_n\Bbb{Z}) \to (\varprojlim X_n) \to \left(\varprojlim X_n/(\Bbb{Z}/b_n\Bbb{Z}) \right) \to 0$

is a short exact sequence of inverse limit systems (provided the second and third systems actually produce inverse limits). I am mostly curious in the fourth inverse limit system (the right-most non-zero inverse limit system), but the documentation I am finding for inverse limits is becoming increasingly incomprehensible for me.