I don’t feel like an all out post, but I did need to use this fact today and it relates to what I’ve been doing, so may as well post the proof. Let be an integrally closed Noetherian domain. Then .

We start with a Lemma: Every prime divisor of a non-zero principal ideal has height 1.

Suppose is a prime divisor of . Then there is an element such that . Define . So . Thus and .

If we had then the standard determinant trick (see a proof a Nakayama’s Lemma) would give us that is integral over contradicting being integrally closed. So and hence and so is a DVR which means dimension of is 1 which means . This gives the lemma.

We will prove that for with we have for all height 1 primes means that is actually in .

Let be the prime divisors of and let be a primary decomposition of where is -primary. But each has height 1 by the Lemma above, so . Thus .

Maybe a short discussion could be useful at this point. The use of this fact came up when trying to prove something about a normal point. This says that at a normal point on a variety, if you want to check if a function regular, you only need to check that it is regular on all codimension 1 subvarieties through that point. This almost immediately gives that if you have a regular function on () where is a normal point. Then the function actually extends to be regular on all of . A function on a dim variety cannot blow up at just a single normal point.

We have defined and used the associated graded ring . Now we want to see how it behaves under completions.

By the last post, we have , so we immediately get that .

A great theorem that I’ll skip proving is that if is Noetherian, and is any ideal, then the completion with respect to the -adic topology is Noetherian. As a corollary we get that for any Noetherian ring , is Noetherian by noting that the completion of the Noetherian ring with respect to the -adic topology is .

After this brief excursion, we’ll come back to the dimension theory we left off from. The next natural place to go is to regular local rings. A local ring is regular if . (Recall that it is always true that ).

Suppose we have a Noetherian local ring such that . Then the following are equivalent definitions of regular: or can be generated by elements.

The first condition implies that , so it implies regular. Regular implies that can be generated by elements, by this post. Lastly, if can be generated by (if you’ve seen the term, this is a system of parameters), then we have a surjective map of graded rings with kernel . So it is an iso. This finishes up the equivalences.

Last time we saw without proof that (for Noetherian local rings) is regular if and only if is regular. Now we can prove it.

By the equivalent definition of regular, is regular iff , but we proved that , so but this happens iff is regular.

We’ll wrap up today with trying to keeping the geometric picture in mind. Regular means non-singular geometrically. So we see that passing to the completion doesn’t introduce any singularities. But since the dimension of the local ring at a point equals the dimension of the variety we actually get that completion of where is non-singular is where is the dimension of the variety.

So if we interpret completion of as the “analytically local” picture, then we see that locally all non-singular points on a variety are analytically isomorphic.

I’ve been fiddling around on here for a few weeks trying to figure out what my next major set of posts should be about. I’ve finally settled. It turns out that algebraic geometry requires knowledge of a ridiculously large amount of commutative algebra. Now I usually try to avoid repeat posting when I know that I’m doing it, but I don’t think I’m going to stick to that rule for this set of posts. For probably at least the next month I’m just going to try to vastly improve my commutative algebra knowledge.

The first topic will be the Hilbert polynomial. The motivation here is that we are looking for some invariants of projective algebraic sets.

Suppose is a graded ring. Then a graded R-module, M, is a module with an abelian group decomposition such that .

Let be a finitely generated graded -module (graded by degree of the polynomial). Then we define the Hilbert function of to be . The function takes as input something from and outputs the dimension of that graded part.

Here is where the Hilbert polynomial enters in. It turns out that actually agrees with a polynomial of degree less than or equal to for large . We will denote this polynomial .

Let’s prove a general fact first. Suppose is defined for all natural numbers. Then if agrees with a polynomial (with rational coefficients) of degree less than or equal to for all , then agrees with a polynomial (with rational coefficients) of degree less than or equal to for all .

Suppose is a polynomial that satisfies the hypothesis of the preceding statement, i.e. for .

Set for and for .

Now just note that for all integers. So we are done since then is a polynomial with rational coefficients of degree less than or equal to .

As you may have guessed, this little fact was to set up an induction for the actual theorem. Let’s induct on the number of variables . The base case just puts us in the case where our graded module is over a field and hence is a finite-dimensional vector space. Thus dimensions all have to be zero at some grading, so for large and we are done.

Suppose the theorem holds in variables. Now let be the kernel of the multiplication map by . This is a submodule of , and we get an exact sequence . Where the means the grading is shifted by .

The exactness tells us something about the dimensions. So look at the part of the grading: . In terms of the Hilbert function, this says precisely that .

Since and are f.g. graded modules over we can apply the inductive hypothesis to the right side. But since the right side is a polynomial for large , so is the left side. Now the fact we proved before this gives us the full result.

There is much to say about Hilbert polynomials, so I’ll probably keep posting about them for awhile.

There was talk about schemes in the comments of my last post, so after reviewing what I’ve already posted about, I decided I may as well package it all up nicely in a brief post so that I’m allowed to use the term freely from now on.

First, recall the sheaf structure we already have. For any ring, R, we have the associated topological space and the sheaf of rings . Then the stalk for is . Also, for any .

Let’s extrapolate what was the important structure here. We really have a topological space and a sheaf of rings on it. We call this a ringed space. Morphism in this category are a pair , where is continuous, and the sheaf structure is preserved, i.e. is a map of sheaves of rings on Y.

A ringed space is called a locally ringed space if each stalk is a local ring. I’m not sure how technical I should be about the definition of a local homomorphism. Essentially, we want to preserve localness on the homomorphisms induced on the stalks by the sheaf homomorphism. So a homomorphism is local if the preimage of the maximal ideal in one go to the maximal ideal in the other.

So without proof I’ll just state that a homomorphism of rings induces a natural morphism of locally ringed spaces (contravariantly), and conversely, given A and B, any morphism of locally ringed spaces is induced by a ring hom . The first statement essentially follows from laying down definitions, but it is not trivial. The second one requires some more thought.

Now we define a scheme. An affine scheme is a locally ringed space that is isomorphic to the spectrum of some ring. A scheme is a locally ringed space in which every point has an open neighborhood such that is an affine scheme. Morphisms are in the locally ringed sense.

The easiest example would be a field, where the topological space is a point and the structure sheaf is the field back again. If we step the dimension up by one (and require the field to be algebraically closed for sake of example), then

I may or may not return to elaborate. I sort of want to consolidate the algebra I’ve learned this quarter through a series of posts before doing anything else along the algebraic geometry side of things.

Wow. I hate looking at the dates on old posts. I think that maybe a few days have gone by, and I’m horrified to find that 11 or 12 days have passed. It is hard to keep track of time in grad school.

The goal of this post is to prove the theorem: If V is an irreducible projective variety over an algebraically closed field k, then every regular function on V is constant. Note this says that . Also, an exercise is to think about how this relates to Liouville’s Theorem if our field is .

Proof: Let V be an irreducible projective variety in . WLOG V is not contained in a hyperplane, since then we could just eliminate a variable and work in and repeat this until it was not in any hyperplane.

Let . Consider the affine covering from last time . Note that is regular as an affine morphism on . So we can write this as a polynomial in where . i.e. we can factor out the homogeneous part of the denominator variable to get where is homogeneous of degree .

But we assumed V irreducible, so is prime and hence is an integral domain. Let’s take the field of fractions then, . Then , and are all embedded in L. So in L we can multiply by that denominator we had before to get .

Recall that is a graded ring, so I just am denoting to be the -graded part. Thus if we take any integer , then is a finite-dimensional k-vector space. Moreover, the monomials of degree N span the space.

Let be a monomial. Then it is divisible by for some i, so . Thus .

So we have a chain: . So for any . Thus .

But is Noetherian, since it is finitely generated as a -module, so is also finitely generated over . Thus is integral over .

i.e. there are such that . But this shows that is homogeneous of degree 0. i.e. . So f is constant.

Last time I recoiled from my statement that projective spaces are manifolds, but let’s explore this a little. We don’t want manifolds, but an idea comes up in showing projective space is one that we can use. The fact that we have natural local “coordinate patches” on projective space. i.e. , where . For a manifold, we would show these are diffeomorphic to copies of , but this isn’t the right topology (and has far too restricting of structure), so we really want to show that each is homeomorphic to affine space by .

So to reiterate, we have the Zariski topology on projective space by closed sets defined as zero sets of homogeneous polynomials, and the Zariski topology on affine space by zero sets of polynomials. What we want to show is that we can cover projective space with affine spaces, so that locally projective space looks affine.

Let’s just work with , since there is nothing special about any of the coverings. As far as sets are concerned, is certainly bijective, so we just need to check that and are continuous. So we’ll check that they are closed maps.

The polynomials that we are concerned with defining closed sets on are from the set and the polynomials defining closed sets on are from .

So now let by , and let by .

It is easy to check that .

Now let be any closed set of . Then where is a closed set in . i.e. is a projective variety, so there is a set such that and .

Any closed set of has the form where and .

But now the image of closed sets under is precisely . So is a closed map. Also, the image of a closed set , so both are closed and hence continuous, so is a homeomorphism.

But this proof actually gives us more. Namely that irreducible projective varieties such that is in 1-1 correspondence with irreducible affine varieties .

I will assume a basic familiarity with projective space from now on (I don’t think Ive covered it in any previous posts). For a quick recap and a guide to my notation, we can define projective space on a finite dimensional vector space V over k by defining a n equivalence relation iff there is a non-zero scalar such that . Then .

Review/facts: . Also, the way we usually think is . The complex projective space (line) is just the compactification of the plane and hence the Riemann sphere. The fact that this is a compact manifold is no coincidence, in fact, all projective spaces can be given the structure of a compact manifold…but maybe this should not be mentioned, since I want to put a different topology on it and talk about varieties.

So our field and the dimension of the vector space are in some sense more important than the vector space itself, so I’ll notate from now on as . If we take as the standard projection onto the quotient, we use the notation for . These are called homogeneous coordinates. Note that these are only well-defined up to scalar multiple.

I haven’t developed on this blog any good motivation to now switch to projective space, but there are some good reasons. At first, it seems to just make things more complicated, but really it simplifies things in the long run. Also, there are some nice properties that you should check. In the affine case, lines can either be parallel and never intersect, or they intersect somewhere. In the projective case, all lines intersect.

So now we just extend the same definitions from the affine case to the projective case, but we are careful to make sure everything is well-defined.

Since homogeneous coordinates are determined up to multiplication by a scalar, we need to make sure our polynomials can deal with this. So we call a polynomial homogeneous of degree d, if every monomial has degree d. i.e. where . So, we have well-defined zero sets of polynomials since we can pull scalar multiples out: , i.e. the variety is well-defined. Note: If d=2, then we call these quadratic forms.

We call a subset a “projective variety” if there is a set of homogeneous polynomials such that .

I’ll let you digest that, and learn a little more about projective space if this was your first experience dabbling with it. Our future plans are to try to figure out which parts of the affine theory we developed carries over unscathed, which parts break down, and which parts get simplified.

Alright, so I’m still taking this really round about way to the Nullstellensatz, but someday I’ll get there.

For those of you that know about sheaves, some of the things I’ve been talking about should be looking vaguely familiar. We haven’t fully gotten there yet, but that is what today is about.

I won’t explicitly define what a general sheaf is, but of course there is always wikipedia or a textbook if you really want to know.

Let’s think back to what we had before. We define what we called the coordinate ring on the algebraic set . So now we do the natural thing, we look at the field of fractions of which we will denote . You should say, “Wait a minute!” at this point, since we might have some “zero denominators.” So let’s hold off on actually defining this until we’ve built the way to work around the problem.

So as a set, is something of the form , where . So it is a fraction of polynomials, or a rational function. The problem is that it is not defined at zeros of . Luckily, zeros of polynomials are all we’ve been studying and talking about for awhile.

Call regular at a point if there is a representation such that . In fact for any we can define a set corresponding to where it can be in the denominator, i.e. . Note that this is just the principal open set we defined earlier for the Zariski topology, but now it seems to have vital use.

Let’s now define the local ring of V at P to be . Clearly this is a subring of . The not as obvious fact is that it is actually local. If you want to check, the unique maximal ideal is the set of elements of the form f/g where and . So now some things are shaping up, since we have an object defined for sets and have a ring of functions at a point.

What would really be exciting is if this construction which seemed ad hoc by taking everything in the field of fractions and throwing out things that don’t work, actually turned out to be a nice localization of the ring. Define the ideal . So this is technically what we were calling before. (The line meaning that we aren’t in anymore, we’re in . So this is is a maximal ideal and hence prime, so we can localize at it.

Exactly what we were hoping for actually does happen, i.e. . In words, the localization of the coordinate ring at .

Now for any open set we define . And for convenience . So not only is a ring, it is a k-algebra. This set of rings with the restrictions we defined last time form the structure sheaf , and the local ring is the stalk of the sheaf at P with the elements as the germ of functions at P.

So I’ll leave you with a nice way to rephrase some older posts: we should now think of , and .

Severely edited: Sorry, some weird bug took out every backslash of this post rendering it incomprehensible. I’m really glad I decided to glance at it randomly.

I’m going to delay the Nullstellensatz for another post, since I’ve touched on a topic, but didn’t really fully lay it out before. This is the notion of an affine variety.

The first thing that should be pointed out is that polynomials are what we care about. Remember that we topologize by saying the closed sets are precisely those collections of points that are the zero set for some collection of polynomials in .

Now let be any closed set. We call a function “regular” if there is a polynomial such that . So a function is regular on a closed set, if it “looks” polynomial on that set.

Now remember that in our ring we have a nice ideal on any closed set . When I say nice, I mean it is quite useful in the information it captures. For instance, it is possible that we have two different polynomials , but that our closed set does not see this, i.e. . But then , which is precisely . Note that all these things were if and only ifs. In some sense, when we are only looking at a closed set, we don’t want to see distinctions that aren’t really there, like polynomials that are the same on the set, but different globally, and now we have captured all of these by this ideal.

We will call the ring of regular functions on V, or the “coordinate ring”. But as we just noted, the way we actually think about this is by .

We have finally come to our definition: An affine variety is an algebraic set V together with its coordinate ring .

This defines a category. We must now specify the morphisms. We’ll say that is a regular morphism if , we have .

So now I think the Nullstellensatz will get some cool results with these definitions in place. I’ll do that next time. For the rest of this time, I should probably list some standard things that are sort of definitional to check:

1) is a regular morphism iff there is some collection such that for all , . i.e. the morphisms are precisely what we’d hope for, just polynomials in each coordinate.

2) Regular morphisms are continuous with respect to the Zariski topology. Again, this is desirable since our category has topological data that should be preserved under morphisms.

An extremely important idea, and much less trivial to prove (though just longer, I don’t think it involves any creative ideas) is that we have a contravariant functor from the category of affine varieties to the category of k-algebras of finite type by and where . It turns out that this is actually stronger than just a mere functor. The two categories are equivalent! We’ll do more on this next time as well, since it turns out that the Nullstellensatz will magically get us all sorts of equivalences between things that are purely geometric with things that are purely algebraic.

I’m also at this point thinking I should add an “algebraic geometry” category so that I stop sticking all these things under “algebra”…

So I have an oops! I try not to repost things other math bloggers have posted, but it turns out my last post was covered plus some here.

I want to move on to some consequences of the Hilbert Nullstellensatz, but it has been covered in many different places:

The result is discussed a little at Secret Blogging Seminar.

Terry Tao proves it in a way I’ve never seen.

Alright, so people can mull this over for a day, and then tomorrow I plan on posting several formulations of the Nullstellensatz and how it pertains to some of the things I’ve been talking about recently.