Geometric meaning of completion and localization

Solution 1:

First of all, localization $R_I$ is only defined if $R\setminus I$ is actually a multiplicative system, i.e. if for $a\notin I$ and $b\notin I$, you have $ab\notin I$. that translates to $ab\in I$ $\Rightarrow$ $a\in I$ or $b\in I$, so $I$ has to be a prime ideal in any case. For intuition, I would advise thinking of $R$ as a finitely generated (and reduced) $k$-algebra, the quotient of the polynomial ring in $n$ variables over an algebraically closed field $k$: Then, $\mathrm{Spec}(R)$ (or at least the maximal ideals among the prime ideals) is just a subvariety of the affine space $k^n$. Think of $R$ as the functions on $X=\mathrm{Spec}(R)$.

1. The ring $R_I$ is the ring where you are allowed to invert any function which is not in $I$: If $I$ is the ideal corresponding to a closed subvariety $Z=Z(I)$ of $X$, this means you may invert anything that does not vanish on $Z$. The intuition is that we think of $R_I$ as the functions that are defined locally around $Z$: If you have any function $f$ defined in a neighbourhood $U$ of $Z$ and it does not vanish on $Z$ itself, then by removing $Z(f)$ from $U$, you still have a neighbourhood of $Z$, but now $f$ is invertible everywhere on $U$. So, if you shrink the support of a function around $Z$ far enough, it is either a unit or it vanishes on $Z$.
   
   Recall that the prime ideals of $R_I$ are precisely the prime ideals of $R$ which do not meet $R\setminus I$, i.e. the ones that are contained in $I$. Those again correspond to subvarieties $Z'$ containing $Z$. So I personally think of $\mathrm{Spec}(R_I)$ as the space that parametrizes the $Z'$.

2. Let me quote Chapter 7 of David Eisenbud's book Commutative Algebra with a view toward algebraic geometry on this one:

   A localization $R_I$ of the affine ring of a variety at the maximal ideal $I$ of a point on the variety represents and reflects the properties of Zariski open neighborhoods of the point; the completion $\hat R_I$ represents the properties of the variety in far smaller neighborhoods. For example, over the complex numbers, the information available from $\hat R_I$ is (roughly speaking) infor- mation about arbitrarily small neighborhoods in the "classical topology" induced by the fact that the variety is a closed subspace of some $\mathbb C$ with its ordinary topology.

   He follows it up with a nice example, I suggest you give it a look.

3. Now this time, we are allowed to invert everything in $(a)$. In other words, you may divide by a function that before may have had zeros. What has happened geometrically? We have removed the zeros of that function from $X$. We think of $R_a$ as the regular functions on the so-called standard open set $D(a)=X\setminus Z(a)$. Again, $\mathrm{Spec}$ passes us from functions to points and we think of $\mathrm{Spec}(R_a)$ as the open subvariety $X\setminus Z(a)$. That is, in fact, very accurate.