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Motivation: In Spanier's Algebraic Topology we are introduced to direct and indirect limits in the first few pages. I partially understand the idea behind it but since I am an example-driven learner I'd actually like to compute some direct limits.

Question: Where can I find explicit examples of a direct (or even indirect) limit being calculated?

What I've done so far: So far, I've computed the direct limit for some finite directed sets. On the wikipedia page, there are a number of examples which I've been trying to work out. In particular, the third example (which is the infinite chain of ${\mathbb Z}/p^{n}{\mathbb Z}\rightarrow {\mathbb Z}/p^{n+1}{\mathbb Z}$ with multiplication by $p$ is the map) I'm unsure of how they come up with the roots of unity for some power of $p$. Any help on this one would also be greatly appreciated.

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up vote 7 down vote accepted

A standard example of a direct limit is the construction of the ring of germs of functions at a point (on a topological space), which generalizes to the stalk of a sheaf.

Consider the germs of holomorphic functions at $0 \in \mathbb{C}$. By definition, an element of this ring is a pair $(U, f)$ where $U$ is an open subset containing the origin and $f: U \to \mathbb{C}$ is a holomorphic function. Two pairs $(U, f), (V, g)$ are equivalent if there is a neighborhood $W \subset U \cap V$ containing $0$ such that $f = g$ on $W$. (We can take $W=U \cap V$ if we make our neighborhoods connected by the rigidity of holomorphic functions.) This is an example of a direct limit: it is the limit of $Hol(U)$ as the neighborhood $U$ shrinks to zero.

It is a good exercise to see that this direct limit is the ring of convergent power series in $\mathbb{C}[[z]]$.

Here is another exercise you might try. Let $A$ be a ring, $S$ a multiplicatively closed subset. Then one can define a preorder (it's not really an order) on the elements of $S$ by $s' \leq s$ if $s'$ divides $s$. Then the localization $A_S$ is the direct limit of the $A_s$ as $s \in S$.

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The first example is quite intuitive! Thank you for that. I'll work out this second example soon. – james Feb 7 '11 at 6:14

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