How do I find the direct limit:
$lim \frac{1}{n}\mathbb{Z}$?
thanks.
How do I find the direct limit:
$lim \frac{1}{n}\mathbb{Z}$?
thanks.
Your question, as stated, doesn't make sense. What are the maps $n^{-1}\mathbb{Z}\to (n+1)^{-1}\mathbb{Z}$? We do get natural maps $n^{-1}\mathbb{Z}\to m^{-1}\mathbb{Z}$ if $n|m$, though (what are they?), and so we can take the colimit of this diagram. You should be able to check that it is isomorphic to $\mathbb{Q}$ because the colimit will identify different ways of writing the same fraction. That is, $\frac{a}{b}\in b^{-1}\mathbb{Z}$ and $\frac{ka}{kb}\in (kb)^{-1}\mathbb{Z}$ will get identiified because the map $b^{-1}\mathbb{Z}\to(kb)^{-1}\mathbb{Z}$ will take $\frac{a}{b}$ to $\frac{ka}{kb}$, meaning they will represent the same element in $\operatorname{colim} n^{-1}\mathbb{Z}$.