Adjectives

Apropos of nothing, I now have a new favorite mathematical term:

“Fake baby monster Lie algebra.”

And I thought “complex simple Lie algebra” was funny.

Localization and the strong Nullstellensatz

A basic idea in topology and analysis is to study a space by restricting attention to arbitrarily small neighborhoods of a point. It is desirable, therefore, to have a notion of looking at small neighborhoods of a point which can be stated in entirely ring-theoretic terms. More generally, we’d like to have a way to ignore some points and only think about others. The tool that allows us to do this is called localization, and it offers a conceptual proof of the strong Nullstellensatz from the weak Nullstellensatz, which, as you’ll recall, is the tool that allows us to describe the category of affine varieties as the opposite of a category of algebras.

MaxSpec is not a functor

For commutative unital C*-algebras and for finitely-generated reduced integral -algebras, we have seen that is a functor which sends homomorphisms to continuous functions. However, this is not true for general commutative rings. What we want is for a ring homomorphism to be sent to a continuous function

via contraction. Unfortunately, the contraction of a maximal ideal is not always a maximal ideal. The issue here is that a maximal ideal of is just a surjective homomorphism where is some field, and the contracted ideal is just the kernel of the homomorphism . However, this homomorphism need no longer be surjective, so it may land in a subring of which may not be a field. For a specific example, consider the inclusion . The ideal is maximal in , but its contraction is the ideal in , which is prime but not maximal.

In other words, if we want to think of ring homomorphisms as continuous functions on spectra, then we cannot work with maximal ideals alone. Prime ideals are more promising: a prime ideal is just a surjective homomorphism where is some integral domain, and the contracted ideal of a prime ideal is always prime because a subring of an integral domain is still an integral domain. Now, therefore, is an appropriate time to replace with , the space of all prime ideals equipped with the Zariski topology, and this time is a legitimate contravariant functor .

In this post we’ll discuss this choice. I should mention that the Secret Blogging Seminar has discussed this point very thoroughly already, but from a much more high-brow perspective.

Affine varieties and regular maps

I have to admit I’ve been using somewhat unconventional definitions. The usual definition of an affine variety is as an irreducible Zariski-closed subset of , affine -space over an algebraically closed field . A generic Zariski-closed subset is usually referred to instead as an algebraic set (although some authors also call these varieties), and the terminology does not apply to non-algebraically closed fields. The additional difficulty that arises in the non-algebraically-closed case is that it’s harder to think about points. For example, has two types of points corresponding to the two types of irreducible polynomials: the usual points on the real line and additional points . These points can be thought of as orbits of the action of on , hence can be thought of as the quotient of by this group action. This picture generalizes.

Anyway, for convenience let’s stick to . In this case, and more generally in the algebraically closed case, there is a reasonably simple description of what the category of affine varieties looks like, but first we have to describe what the morphisms look like and then we have to take the strong Nullstellensatz on faith, since we haven’t proven it yet.

Functoriality

I wanted to talk about the geometric interpretation of localization, but before I do so I should talk more generally about the relationship between ring homomorphisms on the one hand and continuous functions between spectra on the other. This relationship is of utmost importance, for example if we want to have any notion of when two varieties are isomorphic, and so it’s worth describing carefully.

The geometric picture is perhaps clearest in the case where is a compact Hausdorff space and is its ring of functions. From this definition it follows that is a contravariant functor from the category of compact Hausdorff spaces to the category of -algebras (which we are assuming have identities). Explicitly, a continuous function

between compact Hausdorff spaces is sent to an -algebra homomorphism

in the obvious way: a continuous function is sent to a continuous function . The contravariance may look weird if you’re not used to it, but it’s perfectly natural in the case that is an embedding because then one may identify with the restriction of to the image of . This restriction takes the form of a homomorphism whose kernel is the set of functions which are zero on , so it exhibits as a quotient of .

Question: Does every -algebra homomorphism come from a continuous function ?

Textbooks

I recently added two new pages to the blog: a bibliography for listing references I cite on multiple occasions, and a suggestions and requests page. The bibliography is likely to soon contain citations for at least some of the following books which have recently come into my possession:

1. Introduction to the Theory of Computation, Sipser
2. Lectures on Quantum Mechanics, Faddeev, Yakubovskii
3. Representation Theory: a First Course, Fulton, Harris
4. Conceptual Mathematics, Lawvere, Schanuel
5. Concrete Mathematics: a Foundation for Computer Science, Graham, Knuth, Patashnik

I haven’t looked at 2 or 4 very closely yet, but so far I find 1, 3, and 5 to be among the best written textbooks I have ever read. Sipser’s book, in particular, strikes me as having found a perfect balance between brevity and clarity. His tone is conversational but finely polished, and I rather like his habit of summarizing the basic strategy of a proof before actually writing it down. Generally I am finding the book an absolute pleasure to read, which I can’t say for most of the math textbooks I’ve seen. You will likely see me blogging a little about languages and automata once I finish up my current series (right now I’m stuck on what should be a trivial proof).

Why don’t more mathematicians write like Sipser?

The ideal-variety correspondence

I guess I didn’t plan this very well! Instead of completing one series I ended one and am right in the middle of another. Well, I’d really like to continue this series, but seeing as how finals are coming up I probably won’t be able to maintain the one-a-day pace. So I’ll just stop tagging MaBloWriMo.

Let’s summarize the story so far. is a commutative ring, and is the set of maximal ideals of endowed with the Zariski topology, where the sets are a basis for the closed sets. Sometimes we will refer to the closed sets as varieties, although this is mildly misleading. Here denotes an element of , while denotes the corresponding ideal as a subset of ; the difference is more obvious when we’re working with polynomial rings, but it’s good to observe it in general.

We think of elements of as functions on as follows: the “value” of at is just the image of in the residue field , and we say that vanishes at if this image is zero, i.e. if . (As we have seen, in nice cases the residue fields are all the same.)

For any subset the set is an intersection of closed sets and is therefore itself closed, and it is called the variety defined by (although note that we can suppose WLOG that is an ideal). In the other direction, for any subset the set is the ideal of “functions vanishing on ” (again, note that we can suppose WLOG that is closed).

A natural question presents itself.

Question: What is ? What is ?

In other words, how close are to being inverses?

Irreducible components

If it wasn’t clear, in this discussion all rings are assumed commutative.

Given a variety like we’d like to know if there’s a natural way to decompose it into its “components” . These aren’t its connected components in the topological sense, but in any reasonable sense the two parts are unrelated except possibly where they intersect. It turns out that the Noetherian condition is a natural way to answer this question. In fact, we will see that the Noetherian condition allows us to write uniquely as a union of a finite number of “components” which have a natural property that is stronger than connectedness.

The Noetherian condition as compactness

Let’s think more about what an abstract theory of unique factorization of primes has to look like. One fundamental property it has to satisfy is that factorizations should be finite. Another way of saying this is that the process of writing elements as products of other elements (up to units) should end in a finite set of irreducible elements at some point. This condition is clearly not satisfied by sufficiently “large” commutative rings such as , the ring of fractional polynomials.

Since we know we should think about ideals instead of numbers, let’s recast the problem in a different way: because we can write for any , the ascending chain of ideals never terminates. In any reasonable theory of factorization writing and then comparing the ideals , then repeating this process to obtain a chain of ideals , should eventually stabilize at a prime. This leads to the following definition.

Definition: A commutative ring is Noetherian if every ascending chain of ideals stabilizes.

Akhil’s posts at Delta Epsilons here and here describe the basic properties of Noetherian rings well, including the proof of the following.

Hilbert’s Basis Theorem: If is a Noetherian ring, so is .

Today we’ll discuss what the Noetherian condition means in terms of the topology of . The answer turns out to be quite nice.

The weak Nullstellensatz and affine varieties

Hilbert’s Nullstellensatz is a basic but foundational theorem in commutative algebra that has been discussed on the blogosphere repeatedly, but thematically now is the appropriate time to say something about it.

The idea of the weak Nullstellensatz is quite simple: the polynomial ring has evaluation homomorphisms sending for some point , so we can think of it as a ring of functions on . The ideal of functions vanishing at is maximal, so a natural question given our discussion yesterday is whether these exhaust the set of maximal ideals of . It turns out that the answer is “yes,” and there are a lot of ways to prove it. Below I’ll describe the proof presented in Artin, which has the virtue of being quite short but the disadvantage of not generalizing. Then we’ll discuss how the Nullstellensatz allows us to describe the maximal spectra of finitely-generated -algebras.

See also the relevant post at Rigorous Trivialities.