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.

Geometry

The following geometric special case of the more general construction might be a useful introduction. If we restrict ourselves to, say, finitely-generated integral domains over (dual to (irreducible) affine varieties), then a ring can be completely recovered from the ring of regular functions . On the algebraic side we are looking at morphisms . Certainly it’s clear that represents the forgetful functor to since such a morphism is determined freely by the image of , which is an arbitrary element of . But it isn’t obvious where the ring structure on what is a priori just a hom-set

is coming from. That is, it’s not obvious how to define the structure maps in this setting: the addition map , the multiplication map , the additive identity (where is the one-element set), and the multiplicative identity .

In the setting of affine varieties, we see that gets its structure from the fact that is not just a ring but a ring object in the category of affine varieties. This is worth explaining in some detail, so we’ll explain an analogous concept in a situation where there are less structure maps.

Group objects

Let be a category with finite products; in particular, it has a terminal object . A group object in is an object equipped with maps

such that the obvious diagrams commute. These three structure maps are the multiplication, identity, and inverse maps (we also call the second map the unit map), and this is a straightforward generalization of the ordinary definition of a group in . We obtain the definition of a monoid object by ignoring the inverse map.

Example. A group object in is a topological group. Similarly, a group object in the category of smooth manifolds is a Lie group.

Example. A monoid object in the category of monoids is a commutative monoid. This is a form of the Eckmann-Hilton argument. Similarly, a group object in the category of groups is a commutative group.

Example. A monoid object in the category of (small) categories is a monoidal category.

Given any group object in a category, the set acquires the structure of a group as follows. The multiplication map gives rise to a map

via composition, thanks to the universal property of products. Similarly, the identity map gives rise to a map

thanks to the universal property of the terminal object, and the inverse gives rise to a map

.

Since all of these maps are defined via composition from an element of it is fairly clear that all of the relevant diagrams commute. This is the abstract form of “pointwise” multiplication, but it is worth noting that we don’t have to refer to points to give this definition.

It is worth giving a nontrivial example of how this works. Fix a field , not necessarily algebraically closed, and let be the opposite of the category of finitely-generated integral domains over (essentially, affine varieties over ). Let be a group object in (an algebraic group), such as the general linear group (regarded as an affine subvariety of ). Then gives rise to a group

for every finite extension of , the group of -points of . (Any -variety is determined by its -points over all finite extensions : this is a form of the Nullstellensatz. Hence, for example, any real algebraic group is determined by its -points and its -points.)

Cogroup objects

By the Yoneda lemma, giving the structure of a group in a way that is natural in is equivalent to giving the structure of a group object, at least for categories with finite products. It is natural to ask the dual question: what structure does an object need to have in order for to have the structure of a group in a way that is natural in ?

This is the same as asking the original question in the opposite category , so let us assume that has finite products (that is, has finite coproducts). Then it is necessary and sufficient that (as an object in ) is a group object. Dualizing all of the maps, it follows that (as an object in ) is a cogroup object. This is an object equipped with structure maps

where is called the comultiplication (because it is dual to multiplication), is called the counit (where is the initial object), and I’m not sure what people call . In some circumstances it’s called the antipode; perhaps we should go with coinversion.

Many natural categories have no nontrivial cogroup objects. For example, the only cogroup object in is the empty set because nonempty sets have no maps to the initial object (which is the empty set itself). More generally, any category with an initial object and a faithful functor to preserving it has the same property. This includes categories such as .

One of the most important examples of a cogroup object in mathematics is the circle in the homotopy category of pointed topological spaces. The coproduct in this category is the wedge sum , and the initial object (the empty coproduct) is the one-point space, which we’ll confusingly continue to denote by . The comultiplication is the map given by tracing the path of both circles, the counit is unique, and the coinverse is given by tracing the circle in the opposite order. For any pointed topological space with basepoint , is precisely the fundamental group . More generally, the sphere is a cogroup object in , and is the homotopy group . So cogroup objects are not as weird-sounding as they may first appear.

Recovering groups

Consider our original question, but in the simpler situation of the category of groups. Can we recover the underlying set and group operations of a group purely by working in the language of ? The answer is yes. First of all, the forgetful functor is representable: it is given by , since a morphism is freely determined by the image of , which is an arbitrary element of . In this context it makes sense to write as the free group on one generator because this is part of a more general statement: has a left adjoint, the free set functor , hence

and any functor from a category to with a left adjoint is representable by , since

.

Now, giving the structure of a group in a way that is natural in is equivalent to giving a cogroup structure. The coproduct in is the free product, so the comultiplication is given by a map ; it suffices to send the generator of to the product of the generators of , and then it’s not hard to verify that this comultiplication induces the multiplication on . Similarly the counit is the unique map , and the coinverse is the inverse .

More generally, for any set , the hom-set has a natural group structure (the -fold product) giving a cogroup structure on the free group . It is a theorem of Kan that these are the only cogroups in .

Hopf algebras

Before we recover the full ring structure on a commutative ring, let’s try two easier things: recovering the abelian group structure, and recovering the group of units. The answer to the full question will essentially come from combining these, but they are already independently interesting. The forgetful functor sending a ring to its underlying abelian group has the property that its composition is just the usual forgetful functor. has a left adjoint given by sending a set to the free commutative ring on it (a polynomial ring over ), so is representable by . We want to put an abelian group structure on in a way that is natural in to recover , and now we know this is equivalent to putting a cocommutative cogroup structure on .

The coproduct in is the tensor product over , so we want to find a comultiplication . Such a comultiplication is freely determined by , and it’s not hard to see that the correct choice is (dualize this to see what happens). The initial object is , so the counit is , and the coinverse is .

This defines the structure of an affine group scheme on . Since it represents the “additive group” functor, it is called the additive group scheme . An affine group scheme, when viewed in as a cogroup object, is also known as a Hopf algebra (this is a context in which people call coinversion the antipode), and this is one of the many reasons why Hopf algebras are important in mathematics.

The story for the group of units is similar. The composition of the forgetful functor sending a ring to its group of units with is representable by , so we want to make this into a Hopf algebra. This is straightforward: the comultiplication is the diagonal , the counit is , and the coinverse (or antipode) is . The corresponding affine group scheme is the multiplicative group scheme .

Warning: the group operation on a group scheme (as opposed to the group operation on its set of points over a field, say) is more subtle than a set-theoretic map where is the set of prime ideals. While direct product of rings corresponds to disjoint union of prime ideals, tensor product of rings does not correspond to Cartesian product of prime ideals. For example, the tensor product of two fields of different characteristic is the trivial ring, which has empty spectrum.

In other words, the forgetful functor sending an affine scheme to its set of prime ideals preserves coproducts, but does not preserve products. (This implies that it is not representable, and is one limitation of the prime ideal picture of algebraic geometry.) On the other hand, the functor sending a scheme to its set of -points (for any , although the picture is nicest when is a field) is representable, so it does more than preserve products: it preserves all limits. This is a basic reason thinking about -points is nicer than thinking about all the prime ideals at once (Grothendieck’s relative point of view), and why we can get away with thinking geometrically when we work relative to .

Addition and multiplication on the arithmetic plane

The full answer to our original question is now clear. We want to define the structure of a ring object on , which has a definition completely analogous to that of a group object: it’s precisely a group object structure and a monoid object structure, the latter distributing over the former. In this translates into wanting the structure of a coring object on , and thanks to our work above we already know what the structure maps ought to be (just combine the additive and multiplicative group schemes and ignore multiplicative inversion). This generalizes: in the category of affine schemes over an affine scheme (the opposite of the category of -algebras), taking the tensor product with gives a coring structure on with a compatible coaction of (that is, acquires an action of ), and this gives back the -algebra structure on any -algebra. (This is the abstract explanation for the geometric picture we started with, when .)

So we have accomplished our goal: a specific set of structure maps on allows us to completely recover the commutative ring associated to an affine scheme using only the language of the category of affine schemes.

I was mildly surprised when I realized this. is exactly what you expect it to be if is an algebraically closed field, and it’s similar if is a field, but for it’s the arithmetic plane. This is a somewhat more complicated object, and it’s not immediately clear exactly what the function associated to an element looks like in terms of the arithmetic picture.

So let’s work it out. First we’ll deal with a simpler universal object, . It is the initial object in , hence the terminal object in , so there is always a unique morphism . A prime ideal in gives rise to a quotient map into an integral domain, giving a composite (which is already unique). If has characteristic , then is sent to ; otherwise, it’s sent to the generic point . So the unique morphism sorts points in by the characteristic of their residue fields. (Note that sending a point to another point on the geometric side induces a morphism between residue fields in the other direction on the algebraic side.)

Next we’ll talk about in isolation. Recall that consists of the generic point , the prime ideals where is irreducible (not necessarily monic), and the prime ideals where is a prime and is irreducible over . One should think of these as organized by a “prime axis” where projection to the prime axis corresponds to the morphism above. The fiber of this map over is , and the fiber of this map over the generic point is . Motivated by this uniformity, we will refer to as the zero prime and as the characteristic zero prime field ; then we can say uniformly that the fiber over is .

Now let’s turn to the full picture. Given an element of an arbitrary commutative ring, consider the induced morphism . Given a prime ideal of , we get a morphism , and the image of under is as follows. If has characteristic , then lands in . Next, if the image of in satisfies a polynomial relation with coefficients in (which always happens if, for example, is a finite field), then is sent to . (Remember that can be zero here!) Otherwise, it is sent to the generic point of .

So is actually something quite nice: it records all the possible arithmetic behavior of . This is nice if is some ring closely related to such as an “arithmetic curve” where is an irreducible, ideally monic, polynomial. But if is something like the coordinate ring of an affine variety over then there is no arithmetic data and this picture is less useful. (Again, if you are in this situation you should be working relative to anyway.)

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