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## Schanuel’s conjecture and the Mandelbrot Competition

A student I’m tutoring was working unsuccessfully on the following problem from the 2011 Mandelbrot Competition:

Let $a, b$ be positive integers such that $\log_a b = (\log 23)(\log_6 7) + \log_2 3 + \log_6 7$. Find the minimum value of $ab$.

After some tinkering, I concluded that the problem as stated has no solution. I am now almost certain it was printed incorrectly: $\log 23$ should be replaced by $\log_2 3$, and then we can solve the problem as follows:

$\log_a b + 1 = (\log_2 3 + 1)(\log_6 7 + 1) = \log_2 6 \log_6 42 = \log_2 42$.

It follows that $\log_a b = \log_2 21$. Since $a, b$ are positive integers we must have $a \ge 2$, and then it follows that the smallest solution occurs when $a = 2, b = 21$. But what I’d like to discuss, briefly, is the argument showing that the misprinted problem has no solution.

## Constructing Poisson algebras

(Commutative) Poisson algebras are clearly very interesting, so it would be nice to have ways of constructing examples. We know that $k[x, p]$ is a Poisson algebra with bracket uniquely defined by $\{ x, p \} = 1$; this describes a classical particle in one dimension, and is the classical limit of a quantum particle in one dimension (essentially the Weyl algebra).

More generally, if $A, B$ are Poisson algebras, then the tensor product $A \otimes_k B$ can be given a Poisson bracket given by extending

$\displaystyle \{ a_1 \otimes b_1, a_2 \otimes b_2 \} = \{ a_1, a_2 \} \otimes b_1 b_2 + a_2 a_1 \otimes \{ b_1, b_2 \}$

linearly. At least when $A, B$ are unital, this Poisson algebra is the universal Poisson algebra with Poisson maps from $A, B$ such that the images of elements of $A$ Poisson-commute with the images of elements of $B$. In particular, it follows that $k[x_1, p_1, ..., x_n, p_n]$ is a Poisson algebra with the bracket

$\{ x_i, x_j \} = \{ p_i, p_j \} = 0, \{ x_i, p_j \} = \delta_{ij}$.

This describes a classical particle in $n$ dimensions, or $n$ different classical particles in one dimension, and it is the classical limit of a quantum particle in $n$ dimensions, or $n$ different quantum particles in one dimension.

Today we’ll discuss the question of how one might go about constructing Poisson brackets more generally.

## Structures on hom-sets

Suppose I hand you a commutative ring $R$. I stipulate that you are only allowed to work in the language of the category of commutative rings; you can only refer to objects and morphisms. (That means you can’t refer directly to elements of $R$, and you also can’t refer directly to the multiplication or addition maps $R \times R \to R$, since these aren’t morphisms.) Geometrically, I might equivalently say that you are only allowed to work in the language of the category of affine schemes, since the two are dual. Can you recover $R$ as a set, and can you recover the ring operations on $R$?

The answer turns out to be yes. Today we’ll discuss how this works, and along the way we’ll run into some interesting ideas.

## Boolean rings, ultrafilters, and Stone’s representation theorem

Recently, I have begun to appreciate the use of ultrafilters to clean up proofs in certain areas of mathematics. I’d like to talk a little about how this works, but first I’d like to give a hefty amount of motivation for the definition of an ultrafilter.

Terence Tao has already written a great introduction to ultrafilters with an eye towards nonstandard analysis. I’d like to introduce them from a different perspective. Some of the topics below are also covered in these posts by Todd Trimble.

## 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 $\mathbb{C}$-algebra $R$ is dual to the lattice of algebraic subsets of $\text{MaxSpec } R$ (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 $\text{CRing}^{op}$.

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.

## 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 $\mathbb{C}$-algebras, we have seen that $\text{MaxSpec}$ 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 $\phi : R \to S$ to be sent to a continuous function

$M(\phi) : \text{MaxSpec } S \to \text{MaxSpec } R$

via contraction. Unfortunately, the contraction of a maximal ideal is not always a maximal ideal. The issue here is that a maximal ideal of $S$ is just a surjective homomorphism $S \to F$ where $F$ is some field, and the contracted ideal is just the kernel of the homomorphism $R \xrightarrow{\phi} S \to F$. However, this homomorphism need no longer be surjective, so it may land in a subring of $F$ which may not be a field. For a specific example, consider the inclusion $\mathbb{Z} \to \mathbb{Q}$. The ideal $(0)$ is maximal in $\mathbb{Q}$, but its contraction is the ideal $(0)$ in $\mathbb{Z}$, 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 $S \to D$ where $D$ 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 $\text{MaxSpec}$ with $\text{Spec}$, the space of all prime ideals equipped with the Zariski topology, and this time $\text{Spec}$ is a legitimate contravariant functor $\text{CommRing} \to \text{Top}$.

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 $\text{MaxSpec } k[x_1, ... x_n] \simeq \mathbb{A}^n(k)$, affine $n$-space over an algebraically closed field $k$. 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, $\text{MaxSpec } \mathbb{R}[x]$ has two types of points corresponding to the two types of irreducible polynomials: the usual points $(x - a), a \in \mathbb{R}$ on the real line and additional points $(x^2 - 2ax + (a^2 + b^2)), a, b \in \mathbb{R}$. These points can be thought of as orbits of the action of $\text{Gal}(\mathbb{C}/\mathbb{R})$ on $\mathbb{C}$, hence $\text{MaxSpec } \mathbb{R}[x]$ can be thought of as the quotient of $\text{MaxSpec } \mathbb{C}[x]$ by this group action. This picture generalizes.

Anyway, for convenience let’s stick to $k = \mathbb{C}$. 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 $X$ is a compact Hausdorff space and $C(X) = \text{Hom}_{\text{Top}}(X, \mathbb{R})$ is its ring of functions. From this definition it follows that $C$ is a contravariant functor from the category $\text{CHaus}$ of compact Hausdorff spaces to the category $\mathbb{R}\text{-Alg}$ of $\mathbb{R}$-algebras (which we are assuming have identities). Explicitly, a continuous function

$f : X \to Y$

between compact Hausdorff spaces is sent to an $\mathbb{R}$-algebra homomorphism

$C(f) : C(Y) \to C(X)$

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

Question: Does every $\mathbb{R}$-algebra homomorphism $C(Y) \to C(X)$ come from a continuous function $X \to Y$?

## 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. $R$ is a commutative ring, and $X = \text{MaxSpec } R$ is the set of maximal ideals of $R$ endowed with the Zariski topology, where the sets $V(f) = \{ x \in X | f \in m_x \}$ are a basis for the closed sets. Sometimes we will refer to the closed sets as varieties, although this is mildly misleading. Here $x$ denotes an element of $X$, while $m_x$ denotes the corresponding ideal as a subset of $R$; 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 $R$ as functions on $X$ as follows: the “value” of $f$ at $x$ is just the image of $f$ in the residue field $R/m_x$, and we say that $f$ vanishes at $x$ if this image is zero, i.e. if $f \in m_x$. (As we have seen, in nice cases the residue fields are all the same.)

For any subset $J \subseteq R$ the set $V(J) = \{ m | J \subseteq m \}$ is an intersection of closed sets and is therefore itself closed, and it is called the variety defined by $J$ (although note that we can suppose WLOG that $J$ is an ideal). In the other direction, for any subset $V \subseteq X$ the set $I(V) = \{ f | \forall x \in V, f \in m_x \}$ is the ideal of “functions vanishing on $V$” (again, note that we can suppose WLOG that $V$ is closed).

A natural question presents itself.

Question: What is $I(V(-))$? What is $V(I(-))$?

In other words, how close are $I, V$ to being inverses?