Now that I’m settling into the semester, it’s time that I get back to rambling semi-coherently about algebraic geometry. I do, however, now have a somewhat more focused goal, and though it’s still quite a ways off, it has reminded me of important machinery that needs introduction. So for a bit, I’ll be talking more about algebra and some other tools of the trade and putting the geometry itself on the back burner for a bit (which isn’t to say that I’m abandoning it completely).

We begin this project with a tool that I’ve been dancing around for quite some time now, and finally need to sit down and properly define: localization. Before going into intuition for it, we’ll start out right at the definition. Let R be a ring and let S be a subset of R such that 1\in S and x\in S, y\in S\Rightarrow xy\in S. We say that S is multiplicatively closed. We can define a ring S^{-1}R to be pairs (r,s) with r\in R, s\in S with two pairs (a,s),(b,t) equivalent if there exists u\in S such that (at-bs)u=0. Then we perform addition by (a,s)+(b,t)=(at+bs,st) and multiplication by (a,s)(b,t)=(ab,st).

This should be reminiscent of addition and multiplication of fractions, and it is intended to be. We’ll denote (a,s) by a/s in the future.

Now for some examples to see what’s going on. If we take \mathbb{Z} to be the integers and take S=\mathbb{Z}\setminus\{0\}, then S is multiplicatively closed because \mathbb{Z} is an integral domain (ring such that there are no nonzero elements which multiply together to zero. Such elements are called zero divisors.) Then we get S^{-1}\mathbb{Z}=\mathbb{Q}. This is, in fact, the inspirational example for the theory of localization.

Other classical examples are for an integral domain R, we can take the nonzero elements and we obtain a field Frac(R) which is called the field of fractions of R.

More generally, given a ring R and a prime ideal \mathfrak{p}, we can take S to be the elements not in \mathfrak{p}. Then we denote S^{-1}R by R_{\mathfrak{p}} and call it the localization at \mathfrak{p}.

Another example is if we take f\in R and S=\{f^n|f\in \mathbb{N}\}. Then we denote S^{-1}R=R_f. And yes, I know the notation is annoying because subscripts mean two very different things, but that’s how it is.

So now, we can pause for a moment and talk about what these represent in geometry before talking a bit more about the algebra. If R is the coordinate ring of an affine variety (and for reasons we’ll discuss soon, we’ll be able to do most of our work with affines in the future), then R_f is the coordinate ring on the open set given by f\neq 0. Similarly, R_\mathfrak{p} will be the collection of all rational functions which are defined along the zero set of \mathfrak{p}. In particular, localization at a maximal ideal, which corresponds to a point, gives the collection of rational functions defined at that point, we saw this before as \mathscr{O}_x. These are both very important and will come up time and time again in the future.

There is a similar definition of the localization of modules. Let M be an R-module, and S be a multiplicative set. Then there is an S^{-1}R-module S^{-1}M that is the localization of M. We define it to be the set of pairs (m,s) with m\in M, s\in S with the equivalence relation precisely the same as before and addition defined as before.

We will call a statement P a local property if we have M satisfies P is equivalent to M_\mathfrak{p} satisfies P for all prime ideals \mathfrak{p} is equivalent to M_\mathfrak{m} satisfies P for all maximal ideals \mathfrak{m}. LOTS of things are local properties…a module being zero, a homomorphism being injective or surjective, flatness (which we will discuss in the future), and many more things.

Similarly, if we have an operation on rings, modules or ideals such that you get the same thing whether you perform that operation first or whether you localize first, then we say that it commutes with localization. Some examples are finite sums, finite intersections and quotients, taking tensor products, and quite a few other things we’ll encounter. One thing that does NOT commute with localization is tight closure (I won’t be talking about this anytime soon, as my understanding isn’t very good).

About these ads