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Direct limits

  1. Feb 9, 2009 #1
    I'm trying to understand direct limits so consider the direct limit

    [itex]\lim_\rightarrow (\mathbb{Z} \rightarrow \mathbb{Z} \rightarrow \mathbb{Z} \rightarrow \cdots)[/itex]
    where each map is multiplication by 2.
    I concluded that the solution is [itex]\mathbb{Z}\sqcup\coprod(\mathbb{Z}-2\mathbb{Z})[/itex]. Is this correct?
  2. jcsd
  3. Feb 24, 2009 #2
    OK, note first that the direct limit of groups should again be a group.
    In this particular example it is isomorphic to [tex]\mathbb{Z}[1/2][/tex], i.e. the multiplicative group of rational numbers of the form [tex]a/2^b[/tex], where a,b are integers, [tex]b\geq0[/tex].
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