The Picard groups

Let be a commutative ring. From we can construct the category of -modules, which becomes a symmetric monoidal category when equipped with the tensor product of -modules. Now, whenever we have a monoidal operation (for example, the multiplication on a ring), it’s interesting to look at the invertible things with respect to that operation (for example, the group of units of a ring). This suggests the following definition.

Definition: The Picard group of is the group of isomorphism classes of -modules which are invertible with respect to the tensor product.

By invertible we mean the following: for there exists some such that the tensor product is isomorphic to the identity for the tensor product, namely .

In this post we’ll meander through some facts about this Picard group as well as several variants, all of which capture various notions of line bundle on various kinds of spaces (where the above definition captures the notion of a line bundle on the affine scheme ).

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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 be positive integers such that . Find the minimum value of .

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

.

It follows that . Since are positive integers we must have , and then it follows that the smallest solution occurs when . But what I’d like to discuss, briefly, is the argument showing that the misprinted problem has no solution.

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Structures on hom-sets

Suppose I hand you a commutative ring . 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 , and you also can’t refer directly to the multiplication or addition maps , 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 as a set, and can you recover the ring operations on ?

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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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.

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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.

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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.

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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.

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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.

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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 ?

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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?

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