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Archive for the ‘algebraic geometry’ Category

Today’s post is a record of a very small observation from my time at PROMYS this summer. Below, by \text{Spec } R I mean a commutative ring R regarded as an object in the opposite category \text{CRing}^{op}.

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For two categories C, D let D^C denote the functor category, whose objects are functors C \to D and whose morphisms are natural transformations. For C a locally small category, the Yoneda embedding is the functor C \to \text{Set}^{C^{op}} sending an object x \in C to the contravariant functor \text{Hom}(-, x) and sending a morphism x \to y to the natural transformation \text{Hom}(-, x) \to \text{Hom}(-, y) given by composition. The goal of the next few posts is to discuss some standard properties of this embedding and try to gain some intuition about it.

Below, whenever we talk about the Yoneda lemma we implicitly restrict our attention to locally small categories.

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Someone who has just read the previous post on how exponentiating quaternions gives a nice parameterization of \text{SO}(3) might object as follows: “that’s nice and all, but there has to be a general version of this construction for more general Lie groups, right? You can’t always depend on the nice properties of division algebras.” And that someone would be right. Today we’ll begin to describe the appropriate generalization, the exponential map from a Lie algebra to its Lie group. To simplify the exposition, we’ll restrict to the case of matrix groups; that is, nice subgroups of \text{GL}_n(\mathbb{F}) for \mathbb{F} = \mathbb{R} or \mathbb{C}, which will allow us to mostly avoid differential geometry.

The theory of Lie groups and Lie algebras is regarded to be one of the most beautiful in mathematics, and it is also fundamental to many areas, so today’s post is an extended discussion motivating the definition of a Lie algebra. In the next post we will actually do something with them.

For studying the hydrogen atom, our interest in Lie algebras comes from the following. If a Lie group G acts smoothly on a smooth manifold M, its Lie algebra acts by differential operators on the space C^{\infty}(M) of smooth functions, and these differential operators are the “infinitesimal generators” which give us conserved quantities for the evolution of a quantum system on M (in the case that G consists of symmetries of the Hamiltonian). Despite the fact that Lie algebras are commonly sold as a tool for understanding Lie groups, arguably in quantum mechanics the Lie algebra of symmetries of a Hamiltonian is more fundamental. This is important in sitations where the Lie algebra can sometimes exist without an associated Lie group.

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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.

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The following two lemmas might be encountered in a basic course in complex analysis (the first in a basic course in group theory, even).

Lemma 1: Fix a field F. The group of fractional linear transformations PGL_2(F) acts triple transitively on \mathbb{P}^1(F) and the stabilizer of any triplet of distinct points is trivial.

Lemma 2: The group of fractional linear transformations on \mathbb{P}^1(\mathbb{C}) preserving the upper half plane \mathbb{H} = \{ z \in \mathbb{C} | \text{Im}(z) > 0 \} is PSL_2(\mathbb{R}).

I used to only know extremely boring computational proofs of both of these statements. However, I now know better! Today I’d like to give shorter and conceptual proofs of both of these, and then briefly discuss how they come about in the study of elliptic curves (a subject I’d like to talk about in more detail once this semester is over).

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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.

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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.

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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.

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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.

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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?

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