Last time we brought in a bunch of the algebra necessary to do algebraic geometry, now we’ll talk a bit about topology, a bit about morphisms, and then note that we have a perfectly well-defined category (and we’ll even say what that means).
In the first post in this series we defined the Zariski topology in terms of algebraic sets. Now we need to sit down and talk topology a bit more carefully. A topological space is a set along with a collection of sets called open sets such that the empty set is open, is open, the intersection of two open sets is open, and the union of an arbitrary collection of open sets is open.
The classic example is the real numbers with open sets being unions of open intervals (that is, intervals of the form . We’ve mentioned the Zariski topology, and called it a topology. However, the algebraic sets don’t satisfy these axioms. So how can we recover? Well, first, we define a closed set to be the complement of an open set. We note that then it is equivalent to define a topological space as a set with a collection of closed sets, such that are closed, unions of finitely many closed sets are closed, and arbitrary intersections of closed sets are closed. So now we see that the algebraic sets define a topology. We will call the complement of an algebraic set a Zariski open set, or just an open set for short when there isn’t another topology poking around.
In the second post, we mentioned that . In fact, it is a Zariski open subset. This is because, as we defined it, it is . However, an algebraic set is , and so is the complement.
Now with a bit of topology down, we’re ready to go after morphisms of varieties. We’ll go in steps. Let be a variety (all the definitions extend to algebraic sets, but we’ll just talk about varieties for simplicity). Then a function is called regular at a point if there exists an open set containing such that there are polynomials with nowhere zero on such that on . We call a function regular on if it is regular at each point of . To be careful, we should say that we allow arbitrary polynomials for affine varieties, but require homogeneous polynomials of the same degree for projective varieties.
Now we define a morphism from a variety to a variety to be a continuous function (that is, for any open set , the set is open) such that for each regular function , the function is regular for every open.
So now that’s quite a mouthful. The idea is just that morphisms are functions that, when composed with regular functions give us back regular functions.
Before doing any real work with morphisms, let’s quickly detour into categories. A category is a collection of objects along with, for each pair a set of morphisms from to , such that we can compose morphisms in an associative way and such that there is a morphisms called the identity morphism of such that it leaves anything it is composed with unchanged.
Now, I’ll leave it as an exercise (I’ve always wanted to say that) to check that the composition of morphisms of varieties gives another morphism. The composition is associative because composition of functions always is, and these are functions. And also the identity function is a morphism. Thus, we now have a category of varieties, which can be denoted . (the insistence on the will be necessary for later, when we generalize). We say that two varieties are isomorphic if there exist and such that and .
In fact, we can throw a few more things into the category of varieties: the open subsets of varieties. We’ll call an open subset of an affine variety a quasi-affine variety and an open subset of a projective variety a quasi-projective variety. First note that any affine or quasi-affine variety is automatically quasi-projective. And these things are almost as good as actual varieties because they are defined by finitely many conditions of the form or , perfectly nice algebraic conditions. I include these partly for future use and partly because this is standard terminology (as far as I can tell, at least). In the future, we will mostly be talking about projective varieties and quasi-projective varieties. We can get away with that by noticing that an affine variety is an open subset of its projective closure. So all definitions will just be made this way.
Now we’ll define a rational function to be a regular function where is open. We say that it is a rational function . Similarly, rational maps are morphisms defined on an open subset of a variety. Just as the regular functions form a ring (we can add and multiply them pointwise on ) the regular functions on each open set do as well. In fact, if we take all the rational functions and write them as pairs of the open set and the function, and say that if on , then the rational functions form a ring. In fact, a field (that is, a ring in which every nonzero element is invertible).
We call this the function field of the variety, and denote it by . We can in fact get another ring, the local ring of a point by fixing and only looking at the rational functions in which are defined at . We will denote this by . This ring will only have a single maximal ideal, the rational functions which are zero at , and we can also use this ideal to define the tangent space. (We get the same answer as before, essentially.) Similarly, we’ll denote by the ring of regular functions on . So now we’ve got an abundance of rings, and lots of algebra. Now back to geometry:
Let be a morphism. Then we get all sorts of nice maps. By definition of morphism, we have a map which we call the pullback map. This in fact gives us a map . Even better, the map takes the functions that vanish at on to functions that vanish at on . So this even gives us a map . But if we take duals, so that we have tangent spaces, we get a map . This is why we took the dual when we defined tangent spaces yesterday.
We’ve done a lot today, so I’m just going to quote a couple of theorems that are useful without proof. The first is that the set of regular functions on all of a projective variety is just the constant functions. The second is that the function field on a projective (or quasi-projective) variety is just the ratios of homogeneous polynomials in the projective coordinate ring where the numerator and denominator have the same degree. The third and last one is really powerful though, and to show that I’ll point out a corollary. Let be any variety and an affine variety. Then the set of morphisms is in one to one correspondence with the ring homomorphisms .
So now if we assumed that is an affine variety too, then this says that (where the second hom is ring homomorphisms). So now and are isomorphic (written ) if and only if their coordinate rings are isomorphic. To see this, assume that . Then , and so it contains , which is an isomorphism. This corresponds to a map , and also to a map in the opposite direction, and they must compose to the identity, and so the ring homomorphisms are isomorphisms. Now assume that . Then the same argument works in reverse.
The punchline of that theorem is that the affine coordinate ring is the same things as an affine variety. Projective varieties have more structure though.
We’ll leave it at that for now, and if I have time to post tomorrow, we’ll talk about algebraic groups.