The skeleton of Soulé’s F_un geometry

Posted by on Oct 4, 2008 in featured, graduate, Soule2004No comments

The development of a succesful geometry over the nonexistent field with one element  F_1 requires the creation of appropriate categories to work with. So far, we have seen as a part of the  F_1 folklore that we have a nice category of vector spaces over our dearest phantom field, and how this category of vector spaces gives a way to develop an absolute linear algebra. In that linear algebra theory, a fundamental rôle is played by the finite (abelian) field extensions  F_{1^n} .

Christophe Soule tried in his 2004 paper Les varietes sur le corp a un element to take advantage of these field extensions in order to develop a first approach to an algebraic geometry over  F_1 . His main guiding idea is stated right at the begining of the paper:

On part de l’idée qu’une varieté  X (de type fini) sur  F_1 doit avoir une extension des scalaires à  \mathbb{Z} , qui sera un schéma  X_{\mathbb{Z}} de type fini sur  \mathbb{Z}

Which means

We start from the idea that a variety  X (of finite type) over  F_1 must admit an extension of scalars to  \mathbb{Z} , which will be an scheme  X_{\mathbb{Z}} of finite type over  \mathbb{Z}

Since one of our mantras is that we should avoid hitting directly on the elusive  F_1 , trying to work over collections of objects defined over it, the most appropriate reformulation of the notion of scheme that we may use is the, shortest ever, definition (of course equivalent to the classical one):

An scheme over a ground field (or ring)  k is a covariant functor from the category of  k -algebras to the category of sets which is locally representable by an algebra.

With this definition, if we have an scheme  X over  k , given by the functor  \underline{X}: \textrm{Alg}_{k} \to \textrm{Sets} , and  \Omega a k-algebra, the extension of scalars  X \otimes_k \Omega satisfy that for every  k -algebra  R we have an inclusion  X(R) \subset (X \otimes_k \Omega)(R \otimes_k \Omega) satisfying certain universal property. If we want to do the same thing with  F_1 in the place of  k and  \mathbb{Z} in the place of  \Omega , we need to find out at least a big enough collection of rings defined over the field with one element in order to have something to apply our functor! Now, even if it is not very clear what should we take as the category of algebras over  F_1 (I have my own feelings about this, but will ellaborate on that some other day) there is one thing that we can certainly agree: the field extensions  F_{1^n} should be the first candidates to belong to such category.

Now is when wishful thinking comes into play. Since these field extensions are thought of as “adding roots of 1″, the extension of scalars  F_{1^n} \otimes_{F_1} \mathbb{Z} should have rank  n over the integers, and that hints that the following equation should be true:

 F_{1^n} \otimes_{F_1} \mathbb{Z} = \mathbb{Z}[t]/(t^n-1)=: R_n

In particular, for any hypothetical variety  X over  F_1 we must have

 X(F_{1^n}) \subset (X \otimes_{F_1} \mathbb{Z})(R_n)

Going a bit further, we shall assume that the extension of scalars from  F_1 to  \mathbb{Z} induces an equivalence of categories between some (yet to be defined) category of algebras over  F_1 and the category of rings flat and of finite type over  \mathbb{Z} , i.e. rings, whose underlying additive group is a lattice of finite rank.

With this assumption in mind, we have all the ingredients we need in order to start giving a definition of what a variety over  F_1 should be. But that will come in the next post!

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