Optimizing parameters

I came across a fun problem recently that gave me a good opportunity to exercise my approximation muscles.

Problem: Compute , if it exists.

The basic approach to such sums is that the first few terms contribute to the sum because they are large and the rest of the terms contribute to the sum because there are a lot of them, so it makes sense to approximate the two parts of the sum separately. This is an important idea, for example, in certain estimates in functional analysis.

Since it follows that the limit, if it exists, is at least . In fact, this is the precise value of the limit. We’ll show this by giving progressively sharper estimates of the quantity

.

In the discussion that follows I’m going to ignore a lot of error terms to simplify the computations.

Heron’s formula

Heron’s formula for the area of a triangle with side lengths is

where is the semiperimeter. Today I’d like to try to prove this using as little geometry as possible.

Ideals and the category of commutative rings

In this post I’d like to give a better (by which I mean category-theoretic) definition of the lattice of ideals than the standard one. We know that the lattice of ideals has meets and joins defined by intersection and sum, respectively, and that if a lattice is viewed as a category whose arrows are the order relation, then meet and join are the product and coproduct, respectively. But we also know that the lattice of radical ideals of a finitely-generated reduced integral -algebra is dual to the lattice of algebraic subsets of (and that the lattice of prime ideals is dual to the lattice of algebraic subvarieties), and there is a very general category-theoretic formalism for understanding subobjects in a category. It turns out that this formalism reproduces the lattice of ideals of an arbitrary commutative ring – as long as we run it in the opposite category .

Edit, 2/9/10: The above claim is wrong. But let me tell you the construction I had in mind and you can judge whether it is more natural than the usual definition.

Some quadratic reciprocity

In the previous post we showed that the splitting behavior of a rational prime in the ring of cyclotomic integers depends only on the residue class of . This is suggestive enough of quadratic reciprocity that now would be a good time to give a full proof.

The key result is the following fundamental observation.

Proposition: Let be an odd prime. Then contains .

Quadratic reciprocity has a function field version over finite fields which David Speyer explains the geometric meaning of in an old post. While this is very much in line with what we’ve been talking about, it’s a little over my head, so I’ll leave it for the interested reader to peruse.

The arithmetic plane

If you haven’t seen them already, you might want to read John Baez’s week205 and Lieven le Bruyn’s series of posts on the subject of spectra. I especially recommend that you take a look at the picture of to which Lieven le Bruyn links before reading this post. John Baez’s introduction to week205 would probably also have served as a great introduction to this series before I started it:

There’s a widespread impression that number theory is about numbers, but I’d like to correct this, or at least supplement it. A large part of number theory – and by the far the coolest part, in my opinion – is about a strange sort of geometry. I don’t understand it very well, but that won’t prevent me from taking a crack at trying to explain it….

Before we talk about localization again, we need some examples of rings to localize. Recall that our proof of the description of also gives us a description of :

Theorem: consists of the ideals where is irreducible, and the maximal ideals where is prime and is irreducible in .

The upshot is that we can think of the set of primes of a ring of integers , where is a monic irreducible polynomial with integer coefficients, as an “algebraic curve” living in the “plane” , which is exactly what we’ll be doing today. (When isn’t monic, unfortunate things happen which we’ll discuss later.) We’ll then cover the case of actual algebraic curves next.

The connected components functor

I skimmed through books 1, 4, and 5 of my new batch and am currently skimming through 3; it seems I don’t have the mathematical prerequisites to get much out of 2. It will take me a long time to digest all of the interesting things I’ve learned, but I thought I’d discuss an interesting idea coming from Lawvere and Schanuel.

An important idea in mathematics is to reduce an object into its “connected components.” This has various meanings depending on context; it is perhaps clearest in the categories and , and also has a sensible meaning in, for example, for a group . Lawvere and Schanuel suggest the following way to understand several of the examples that occur in practice:

Let be a concrete category with a forgetful functor . If it exists, let be the left adjoint to . Then describes the “discrete” (i.e. “totally disconnected”) objects of , and, if it exists, the left adjoint to is a functor describing the “connected components” of an object in .

I think this is a nice illustration of a construction that is illuminated by the abstract approach, so I’ll briefly describe how this works for a few of my favorite categories.

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.