Motivation for direct limit and filtered directed limit?

Question

I am studying the concept of direct limit and filtered directed limit am wondering what sort of theorems they are used to prove. I know that $\mathbb{Q}/\mathbb{Z}$ can be represented by filtered directed limit. Can anybody please give more detailed examples of how this concepts can be useful, in the setting of algebraic geometry/number theory?

Answer 1

The more general concept of colimits and limits which encompasses filtered limits, inverse limits and all the rest of the limits is very very useful. We can for example define what the kernel and the image of a morphism in a general category is. This allows us to define what an exact sequence of objects $$...\rightarrow A \rightarrow B \rightarrow C \rightarrow ...$$

means in a general category. It is exact when the image of any morphism is equal to the kernel of the one after it. We can talk about exact sequences of sheaves, presheaves, pointed topological spaces or $\mathscr O_X$-modules or any set of nice enough objects you want!

You can also obtain new schemes from gluing other schemes together. This action of gluing is formalized using colimits. Exercise 2.12 in Hartshorne can be rephrased as showing that X is the colimit of the diagram consisting of all the $\phi$ and inclusions.

A more number theoretic example of how limits are used is how we define the ring of p-adic integers $\mathbb Z _p$. They are defined as the inverse limit of the rings $\mathbb Z/p^k\mathbb Z$ with morphisms given by the $\mathbb Z/p^k\mathbb Z \rightarrow \mathbb Z/p^l\mathbb Z$ induced by the inclusion $p^k\mathbb Z \hookrightarrow p^l\mathbb Z$ if $l \leq k$.