The Structure Sheaf of a Variety

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 k[V] the coordinate ring on the algebraic set V. So now we do the natural thing, we look at the field of fractions of k[V] which we will denote k(V). 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, f \in k(V) is something of the form f=g/h, where g, h \in k[V]. So it is a fraction of polynomials, or a rational function. The problem is that it is not defined at zeros of h. Luckily, zeros of polynomials are all we’ve been studying and talking about for awhile.

Call f \in k(V) regular at a point P \in V if there is a representation f=g/h such that h(P) \neq 0. In fact for any h \in k[V] we can define a set corresponding to where it can be in the denominator, i.e. V_h=\{P \in V : h(P) \neq 0\}. 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 \mathcal{O}_{V, P}=\{f \in k(V) : \ f \ regular \ at \ P\}. Clearly this is a subring of k(V). 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 f(P)=0 and g(P) \neq 0. 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 \overline{M}_P=\{ f \in k[V] : f(P)=0\}. So this is technically what we were calling \overline{I({P})} before. (The line meaning that we aren’t in k[x_1, \ldots, x_n] anymore, we’re in k[V]=k[x_1, ldots , x_n]/I(V). 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. k[V]_{\overline{M}_P}=\mathcal{O}_{V, P}. In words, the localization of the coordinate ring at \overline{M}_P.

Now for any open set U \subset V we define \mathcal{O}(U)=\{ f \in k(V) : f \ regular \ on \ U\}. And for convenience \mathcal{O}_V(\emptyset)={0}. So not only is \mathcal{O}_V(U) a ring, it is a k-algebra. This set of rings with the restrictions we defined last time form the structure sheaf \mathcal{O}_V, and the local ring \mathcal{O}_{V, P} 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 k[V]=\mathcal{O}(V), and \mathcal{O}(V_h)=k[V][h^{-1}]=k[V]_h.

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.


Blogging Birthday

April 28…hmmm…why does that day sound familiar? Oh yes, that’s right, it is the one year anniversary of my wordpress blogging! So I guess the cliche thing to do is to post my five favorite posts and my five least favorite posts.

5) I guess I should include Abduction as Logical Inference just because it is one of the most active and read posts of my blog. I’m not as fond of it on this re-reading, though.

4) My favorite sequence of posts has to be when I went through the book Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being almost chapter-by-chapter breaking apart and presenting my ideas about this radical way of thinking about math. The first in the series starts here

3) My letter to Sam Harris has to make the list. I think Sam Harris is absolutely fantastic, and almost no one has influenced my thinking as much as him, but I still think I’ve found a slight flaw in his argument. Maybe it can be patched up. Maybe I’m misinterpreting him a little. Either way, I wish he’d let me know. I actually sent a modified version of this letter to him and got no reply.

2) Posting on “Lost in the Funhouse” was probably the best thing I could do for traffic to my blog. Unbeknownst to me, it seems that everyone who reads this book for school is scouring the internet for some help. Anyway. It was a fun post, and now I realize that I promised a second post about it and it never came. Oh well.

1) And the winner is…of course, my series of posts on the meaning behind my blogging name “hilbert theorem 90”. Here is a link to the most general version of the theorem that I posted. The sequence of posts actually starts here. This was one of the other very clearly most active posts.

Worst things I did and must remember to avoid for the next year:

5) Avoid titles that could come up in a search for the opposite thing. My post “When Art is Meaningless” has brought in so many hits for “art is meaningless” that I am actually the top google hit for that search phrase.

4) I should try to avoid posting on things I know nothing about. Er uhm…my 4 post long attempt at noncommutative geometry

3) Random speculation should be kept to a minimum. I’m scared to think about how many people took this post to have some merit (just based on search hits, it has got to be more than just a couple).

2) I have sort of a toss up when it comes to what my worst decision was. So in second place I’m going to put “heat of the moment” posts. The three that immediately come to mind were all posted after really heated debates. On parenting is sort of along the lines of The Ethics of Manners in which I conclude that having good manners is unethical. And the third one being Ethical Voting Habits which I at least think had quite a bit of merit, but probably should have been posted under a cooler head.

1) The winner for worst decision of my first blogging year was what I called SSS, for Strange Sunday Specials. The first one is here. For those that weren’t following at the time, I tried to post a random fact I found on wikipedia using the random page generator that somehow related to a topic in my blog (sub)title. The attempt was to keep me posting at least once a week during a pretty busy point in my year. It sort of worked, but only to the dismay of my readers who had to read dumb randomness.

Well. I hope you enjoyed the year as much as me. It has been great fun so far and I don’t plan on stopping anytime soon.

Affine Varieties

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 \mathbb{A}^n by saying the closed sets are precisely those collections of points that are the zero set for some collection of polynomials in k[x_1, \ldots, x_n].

Now let V\subset \mathbb{A}^n be any closed set. We call a function f:V\to k “regular” if there is a polynomial F(x)\in k[x_1, \ldots , x_n] such that f=F\Big|_{V}. 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 I(V)=\{f : f(v)=0  \forall v \in V\}. 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 f, g\in k[x_1, \ldots, x_n], but that our closed set does not see this, i.e. f\big|_{V}=g\big|_{V}. But then (f-g)\Big|_{V}=0, which is precisely f-g\in I(V). 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 k[V] 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 k[V]=k[x_1, \ldots, x_n]/I(V).

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

This defines a category. We must now specify the morphisms. We’ll say that \phi: V\to W is a regular morphism if \forall f\in k[W], we have f\circ \phi \in k[V].

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) \phi : V\to W is a regular morphism iff there is some collection F_1, \ldots, F_m\in k[x_1, \ldots, x_n] such that for all v\in V, \phi(v)=(F_1(v), \ldots, F_m(v)). 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 V\mapsto k[V] and \phi\mapsto \phi^* where \phi^*(f)=f\circ \phi. 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”…

Hilbert Nullstellensatz

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.

Some Notions of Dimension

It is time to pull together some ideas we’ve built up, and show that they actually correlate how we want them to.

Recall that we have a notion of dimension for a ring called the Krull dimension. Review this if necessary, but essentially you just take the sup of the heights of all prime ideals in your ring.

For a topological space, we first define “irreducible.” Irreducible simply means that you can’t express the space as the union of two nonempty closed sets. To familiarize yourself with this definition some more you can try to prove that it is equivalent to the definition that any two non-empty open sets intersect non-trivially, or any non-empty open set is dense. So note right away that irreducible is a pretty rough condition. Almost none of the spaces I usually talk about are irreducible, since there are tons of non-dense open sets. And by the other criterion, any Hausdorff space is reducible.

Moving on, we now can define the dimension of a topological space to be sup\{n : Z_0\supsetneq Z_1\supsetneq \cdots \supsetneq Z_n\} where the Z_i‘s are irreducible subspaces.

Naturally, we are now interested to see if the topological notion of dimension on the topological space Spec(R) is the same as the Krull dimension of the ring R.

First, we’ll need a quick lemma:

A subspace E\subset Spec(R) is irreducible if and only if I(E) is a prime ideal. Note that this is really exactly what we wanted to happen, since prime ideals are points in Spec(R). When we say something is “irreducible,” what we are talking about are the smallest things that cannot be broken apart, i.e. points. I confess I am far oversimplifying this idea of “points” as we will see before the week is over if all goes as planned. (At this point you might want to review spec).

Proof of Lemma: I promise to fill this in later in the week. I just realized that it is an assignment to be turned in on Friday, and not all the readers of this blog that are in my commutative algebra class have this done yet.

Now let’s actually check dimensions. Theorem: Krulldim(R)=dim(Spec(R)). Just write it out now:

dim(Spec(R))=sup\{n: Z_0\supsetneq Z_1\supsetneq \cdots \supsetneq Z_n , \ Z_i \ irreducible\}
=sup\{n: p_0\subsetneq p_1\subsetneq \cdots \subsetneq p_n  , \ p_i=I(Z_i)\}
= Krulldim(R) where that switching comes from the Lemma. The correspondence is 1-1.

So for some future posts, I want to clarify some more on dimension and what “points” really are. I also want to do the Nullstellensatz and talk about why it is so important, but I may not. I’ll do some hunting to see what other math bloggers have done on it. I’m pretty sure it has been posted on extensively already, in which I’ll just point people in those directions and add some things that I personally find interesting.

Amazon controversy

Just in case any of my readers aren’t addicted to the internet, and hence have not come across it yet, there is a big controversy going on about I think it fits the realm of my topics, since it has to do with censorship and literature. I was waiting on this post to see if any new non-hearsay information would be released, but it is taking too long, so….

Amazon has decided to label a whole bunch of LGBT (lesbian, gay, bisexual, and transgender) literature (and I think films as well) as “adult”. Which means they remove the ranking. This may not seem all that important, except the ranking is used in the search function. So now horrifically incorrect things pop up if you search for certain books, and some don’t come up at all.

Now this doesn’t seem like all that bad an idea if you are the type of person that thinks adult material should not pop up when children enter search terms into Amazon, except there is a very clear bias and censorship going on. Very explicit heterosexual material has not been affected as well as some sex toys, yet literature that is either non-fiction about gay people or fiction with subtle gay themes and no sexual scenes at all have been censored out.

Anyway, I don’t care to post on this forever since any quick search will bring up lots of people who know way more about the situation than I, and I need to get back to homework rather than research what is really going on. Just thought I’d keep people up-to-date that don’t have their google reader bombarded with hundreds of items a day.

Music Update

I had a large sequence of heavy math posts before the past couple trivial ones. So since I plan on heavy math coming up soon, I’ll alleviate this tedium for my non-math readers. I haven’t posted on music since January with Andrew Bird and Duncan Sheik if I remember correctly, so it is about time to do that again.

I haven’t been keeping incredibly up-to-date in my music this year. I do have to say that even though Joanna Newsom has popped up on my radar several times, I never actually got around to listening to her until recently. I got the (in)famous Ys. I think it is fantastic. For those who don’t know, she is a harpist/songwriter. Her songs (at least on this album) are quite long (some close to 20 minutes). They are also through composed (no verse/chorus repetitions). This is sort of an ancient song form from when bards recounted entire histories in a song. I love the idea and wish it were used more. Also, she is quite experimental in style. The harp is prominent, but most use a full orchestra. The time signatures are very obscure and change rapidly. It is often not in a single key, with chords coming from all over. Yet the whole thing seems very fluent and never unnatural. Everything I look for in music.

My next pick is Loney, Dear. He toured with Andrew Bird (possibly still is), which is why I checked him out. It was exciting at first. He has lots of energy and cool ideas. But the songs grew old quickly for me, and I consider it just an OK album. I’ll probably return a few times when I want some catchy upbeat stuff, but not a lot.

The really surprising one is Doves. I’ve been a faithful follower of the Doves for a long time. I have all their previous albums and like them a lot. I’ve always considered them good, but I think this album is actually great. It is truly an artistic step up from their previous stuff. The influences range from all over the place from blues to electronica/dance. They took an incredibly long time to put this out, and it is very noticeable. The attention to detail is impeccable. I never stop noticing all sorts of little things all over the place. The faster more electronic type songs build constantly in energy and intensity to great climatic moments. The more down-tempo songs have such a great blues or jazz feel that you would never expect from a rock band. Oh yeah, they are mostly a rock band, these are subtle influences I’m talking about.

Lastly, we’ll go local. I heard this guy Zach Harjo on the local radio station and was blown away. Fortunately two days later he was playing a concert a few blocks away from me. I attended and was even more blown away. He has an album that doesn’t seem to have released yet. I talked about Doves being good at infusing their rock with blues, but this is a different blend with the same idea. It is more like infusing blues with rock. Their live show with actual solos is fantastic. This band truly has the feel and sound internalized. It was quite a moving experience (which most people who have experienced really good live blues will probably tell you). I’m looking forward to seeing more from him, especially since he lives near me.