What is the importance of localization in algebraic geometry?

Question

It is often asserted in commutative algebra texts that localization is important in algebraic geometry. I would appreciate some precise examples which show the utility of the concept in this context. Although I have some knowledge of algebraic geometry, I am not an expert, so I would like the examples to be relatively simple if possible.

Answer 1

One of the philosophical points of algebraic geometry is to study not the geometric object itself, but the set of functions on it. Imagine the easiest geometric object: the line. Now, we must work over some field $K$ so the line is $K$. What is a natural ``function space'' on $K$? Well, as we are working with algebraic stuff our "functions'' are polynomials. From the algebro-geometric viewpoint the line IS the ring $K[x]$. To the point on the line one can associate an ideal of functions vanishing in that point. In our case it would be $(x-a)$. So we have the "space'' $K[x]$ and the "point" $(x-a)$. One can ask, what is the space $K[x]$ near the point $(x-a)$? I.e. how the "infinitely" small neighborhood of this point is structured? The functions on a small neighborhood of $a$ are just functions that are not zero near $a$. And of the function is not zero near $a$ then the inverse function is still "regular". So the "functions'' on this small neighborhood are exactly the ring $K[x]_{(x-a)}$.

Don't know whether this makes sense, maybe it will be easier just to properly read some introduction to algebraic geometry.

P.S. We can work not over a field, but i think that is a situation where geometric intuition can be applied.