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Thread: Inverse Limit

  1. #1
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    Inverse Limit

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

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

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

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

    $\displaystyle \begin{matrix} X & \stackrel{\phi}{\longrightarrow} & B \\ \downarrow & & \downarrow \\ X_n & \stackrel{\phi_n}{\longrightarrow} & \Bbb{Z}/b_n\Bbb{Z}\end{matrix}$
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  2. #2
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    Re: Inverse Limit

    I guess I should ask does the map $\displaystyle X \to X_n$ even make sense? I mean, $\displaystyle X$ is essentially an infinite-tuple, and the map would require taking the last $\displaystyle n$ positions... but that might not even be possible. Perhaps the inverse limit of $\displaystyle X_n$ doesn't even exist...
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  3. #3
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    Re: Inverse Limit

    An easier example:

    Define a map from $\displaystyle \Bbb{Z}^m \to \Bbb{Z}^n$ for all $\displaystyle m > n$ by $\displaystyle (x_1,\ldots,x_m) \mapsto (x_{m-n+1},\ldots, x_m)$. Does $\displaystyle \varprojlim \Bbb{Z}^n$ exist given those maps?
    Last edited by SlipEternal; May 29th 2014 at 05:25 PM.
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  4. #4
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    Re: Inverse Limit

    Oh, and for my first example, I have

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

    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.
    Last edited by SlipEternal; May 30th 2014 at 04:00 AM.
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