Let $R$ be a commutative ring and $x\in R$ be a nonzero divisor. Then i know that the direct limit of $R\mapsto R\mapsto R\mapsto\cdots $, where each map is multiplication by $x$ is $R_x$, the localization of $R$ at ${1,x, x^2,…}$.
Similarly the direct limit of $R/x^n\mapsto R/x^{n+1}\mapsto\cdots $, where maps are multiplication by $x$ is $R_x/R$.
My Question: How does one guess what the direct limit of a given direct system is, once the guess is made, then one can go about proving it using the universal property. Can someone provide an intuition for direct limits, at least in the above 2 cases? Thanks
Best Answer
I think the touchstone for understanding direct limits is understanding directed unions.
A collection $C$ of sets is directed if for every $X,Y\in C$, there exists $Z\in C$ containing both $X$ and $Y$. This becomes a direct system using inclusion mappings.
Now just by using the directness of this collection, you can compare any two sets (and inductively, any finite number of sets) by finding a set that contains them all. But what if you want to compare more than finitely many? That is what the limit is going to do: the direct limit for the system above turns out to be $\cup C$, and so you get a sense that the limit is "the limit of finite approximations by compositions of the morphisms".