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Thread: Confusion about Direct Limit of R-Modules

  1. #1
    Senior Member slevvio's Avatar
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    Confusion about Direct Limit of R-Modules

    Hello, I just have a quick question about direct limits. Let$\displaystyle \{ A_s | s \in S \}$ be a directed system of $\displaystyle R$-modules, where $\displaystyle S$ is a directed set. I want to show that

    $\displaystyle \displaystyle\lim_{\longrightarrow} A_s := \displaystle\coprod_{s\in S} A_s/ \sim$

    is an $\displaystyle R$-module, where $\displaystyle a \in A_s \sim b \in A_t$ if there exists $\displaystyle u \ge s,t$ such that $\displaystyle f_{su}(a) = f_{tu}(b) \in A_u$.

    Now in my notes we define addition as follows:

    The sum of $\displaystyle a \in A_s$ and $\displaystyle b \in A_t$ is given by choosing an element $\displaystyle u \in S$ with $\displaystyle u \ge s,t$ and $\displaystyle a+b := f_{su}(a) + f_{tu}(b)$ in the $\displaystyle R$-module $\displaystyle A_u$.

    But my question is, surely this thing is not well defined because there might be some $\displaystyle u' \ge s,t$, where $\displaystyle u' \not= u $! So this addition would be in a different $\displaystyle R$-module. Does this limit module only make sense if in our directed set $\displaystyle S$ we make a specific choice of element $\displaystyle u $ for every $\displaystyle (s,t)$ pair in $\displaystyle S$?

    Thanks for any help.
    Last edited by slevvio; Nov 4th 2011 at 01:48 PM.
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  2. #2
    Senior Member slevvio's Avatar
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    Re: Confusion about Direct Limit of R-Modules

    Just after posting this I think I've worked it out. If we supposed we have $\displaystyle u' $ such that $\displaystyle u' \ge s,t$, then choose $\displaystyle T \ge u,u' \ge s,t$. Then $\displaystyle f_{uT}(f_{su}(a)) = f_{sT}(a) = f_{u' T}(f_{su'}(a))$. Therefore $\displaystyle f_{su}(a) \in A_u \sim f_{su'}(a) \in A_{u'}$. By a similar argument we have $\displaystyle f_{tu}(b) \in A_u \sim f_{tu'}(b) \in A_{u'}$.

    Therefore in the direct limit, $\displaystyle a+b$ is the same no matter which $\displaystyle u$ we pick. $\displaystyle \Box$
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