Gadgets a la Soulé

We already stated the guiduing principle to Soulé’s geometry over F_un; namely, that the notion of variety over the field with one element ought to be well bahaved under extension of scalars to the ring of integers .

We also mentioned the fancy Grothendiecksque definition of a scheme over as a (locally) representable functor from -algebras to sets. If you are one of the many people who are more used to think in geometrical terms rather that using a fanshi-wanshi categorical interpretation this might sound scary, but the idea behind it is actually quite simple: given a scheme over (defined in any classical way you feel comfortable with) we can associate to it the functor

that assigns to every (commutative) algebra the set of morphisms of schemes between the spectrum of and , which is what is normally called the set of -points of .This kind of construction is sort of standard in abstract algebraic geometry, and often receives the name of the functor of points. In the simplest case, when we are dealing with an affine scheme, i.e. something given as , then the -points of are simply the ring homomorphisms from to . Now, this gives us a functor associated to our scheme, but, is this a good correspondence? In other words, can we recover all the structure of the scheme just using the functor of points? The answer to this question is yes, we can, and the reason lies on a subtle but important fact: we have enough objects in the source category (i.e. enough commutative algebras) so that the scheme is fully determined by its points on every single algebra. For practical purposes, we might replace the category of all algebras by something smaller but still rich enough, and that would still give us an equivalent definition of scheme. This will come handy when we try to extend this definition to the case of varieties over . If all this sounds too complicated, you should maybe start by reading this other post first before trying to go on.

So, for Soulé, a gadget (for lack of a better translation of the French ‘truc‘)  over consists on a couple , where is a (covariant) functor defined on a suitable category and landing on sets, and is a (complex) algebra. The functor should be the analogous to the functor of points given by a cassical variety; tricky point now is the choice of the category , which would be the substitute of the category of algebras. Different choices here will lead to different versions of geometries. If our category is too big we might be in trouble to find any examples of functors, and if it is too small it might happen that the functorial information is not enough to recover a geometrical object. There is no canonical choice here. Since our base test object are the field extensions of , the category should contain at least the rings , and if we ever want to be able to define a nice cartesian product of varieties, we’d better drop in their tensor products as well. This would give us a minimal choice for our base category. A richer choice, which is indeed the one chosen by Soulé in his paper is taking the category of all (commutative, unital) rings that are flat and finitely generated over the integers (recall that every ring is a module over in a canonical way). These conditions, on a ring , just mean that the underlying additive group of is a lattice of finite rank. Examples of such kind of rings are for instance all the rings of integers of number fields, such like , and similars. The rings are of course of this type, since they are the rings of integers of cyclotomic extensions of the rationals.

As we already mentioned, the choice for this category for defining gadgets can be pushed further. One possible refinement, undertaken by Connes and Consani, and explained here, is not taking rings but abelian groups instead. Since every ring is also an abelian group, this produces quite an enlargement of our testing category, and thus a way more restrictive notion of geometry.

But for now, we shall stick to our category of nice rings. So, as we were saying, a gadget is given by a functor , plus an algebra . We will also require that for any ring morphism and every element we have a way of “evaluating” elements of the algebra in , like if were an algebra of functions (which will often be the case). More concretely, we will require the existence of algebra homomorphisms , and that these evaluation maps behave well with respect to morphims in , meaning that whenever we have a morphism , for any point we have the equality .

The rôle of the algebra is a bit mysterious at the moment. The reason for introducing it is that, unlike in the classical case, the functor of points is not enough to determine the extension of scalars to of a hypothetical variety over . Morally, this algebra will represent a “given topology at infinity”, in the spirit of Arakelov geometry. The properties that this algebra ought to satisfy are not clear at the moment, so we will be quite vague about them.

Now that we have a collection of objects, our gadgets, we need an appropriate notion of morphism between gadgets. For gadgets and , a morphism will consist on a couple , where is a natural transformation of functors, and is an algebra morphism such that, for each ring , algebra map, and each we have the equality as maps from to .

Composition of two morphisms and is simply given by .

All the above added up together, we have constructed a category, called the category of gadgets over . This is not yet what we need in order to have a variety with the properties we expect, so we will have to put some restrictions (related to the ‘local representability’ of the functor of points) in the definition… but we will save that for the next time!