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Math Help - 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  \{ A_s | s \in S \} be a directed system of R-modules, where S is a directed set. I want to show that

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

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

    Now in my notes we define addition as follows:

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

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

    Thanks for any help.
    Last edited by slevvio; November 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 u' such that u' \ge s,t, then choose T \ge u,u' \ge s,t. Then f_{uT}(f_{su}(a)) = f_{sT}(a) = f_{u' T}(f_{su'}(a)). Therefore f_{su}(a) \in A_u \sim f_{su'}(a) \in A_{u'}. By a similar argument we have f_{tu}(b) \in A_u \sim f_{tu'}(b) \in A_{u'}.

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