# Motivation for rings of fractions?

I'm learning about rings of fractions and localization. I like the material a lot and feel engaged with it, but I do lack a broader perspective on things. For example, I'm aware of things such as $S^{-1}A$ is the "best" ring for making everything in $S$ a unit (via the universal property), or that localization at a prime ideal gives us a local ring.

These are certainly nice things to have, but I'm wondering if there's a relatively simple example in algebraic geometry or number theory which will demonstrate the usefulness of these notions.

(In particular, a common comment I don't understand yet is along the lines of how constructing rings of fractions is analogous to "concentrating attention to an open subset or near a point." My understanding of algebraic geometry is still very vague at this point and includes only very basic facts about varieties and their correspondence with ideals.)

The basic building block in that theory is the affine scheme $X=\operatorname {Spec}(A)$ associated to an arbitrary commutative ring $A$.
The scheme $X$ comes with a topology, a canonical basis $(D(f))_{f\in A}$for that topology and, above all, with a sheaf of rings $\mathcal O_X$ which associates to every $D(f)$ the ring of sections $\Gamma(D(f),\mathcal O_X)=A_f$ .
The relevance to your question is that the above ring is the ring of fractions $A_f:= S_f^{-1}A$ associated to the multiplicative monoid $S_f= \{1,f,f^2,f^3,\cdots\}$.
Moreover the points of the topological space $X$ are the prime ideals $\mathfrak p\subset A$ and the stalk $\mathcal O_{X,\mathfrak p}$ of $\mathcal O_X$ at $\mathfrak p$ is the ring of fractions $A_\mathfrak p=S_\mathfrak p ^{-1}A$, where $S_\mathfrak p$ is the multiplicative monoid $S_\mathfrak p=A\setminus \mathfrak p$.
• Dear @Manos, the ring of regular functions on $U$ is just $k[x,y]$ and not the localization of that ring of polynomials at anything: see here. What you might be missing is that $U$ is not of the form of $D(f)$ for any $f\in k[x,y]$: for example $D(xy)\subsetneq U$ is much smaller than $U$. – Georges Elencwajg Apr 16 '16 at 17:20
A "relatively simple example in number theory" is the following. The ring $A=\mathbb{Z}$ localised at the ideal generated by a prime $p$, with $S=A\setminus (p)$, gives the ring $\mathbb{Z}_{(p)}$ of $p$-local numbers, whose completion are the integer $p$-adic numbers $\mathbb{Z}_{p}$. It's field of fractions are the $p$-adic numbers $\mathbb{Q}_p$. These are important rings and fields in number theory.
Consider the affine space $\mathbb{A}^2$. It can be shown that the regular functions on $\mathbb{A}^2$ are the polynomials $k[x,y]$. Now consider the curve $Y$ of $\mathbb{A}^2$ given by the equation $x^2-y = 0$. Suppose we are not interested at all in this curve and we want to remove it from our space, i.e., we want to work inside the open set $U = \mathbb{A}^2 - Y$. It is reasonable to want to know what are the regular functions on our open $U$. As one may guess, the regular functions on $U$ are the regular functions on $\mathbb{A}^2$, but are now "enhanced", in the sense that we can divide any element of $k[x,y]$ with any power of the polynomial $x^2-y$: since we are working outside $Y$ this is a valid operation. Formally, the ring of regular functions on $U$ is the localization of $k[x,y]$ at the element $x^2-y$.