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The skeleton of SoulÃ©’s F_un geometry

Posted By __javier__ On October 4, 2008 @ 6:23 pm In __featured,graduate,Soule2004__ | __No Comments__

The development of a succesful geometry over the nonexistent field with one element requires the creation of appropriate categories to work with. So far, we have seen as a part of the folklore ^{[1]} 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 ^{[2]}. In that linear algebra theory, a fundamental rÃ´le is played by the finite (abelian) field extensions .

Christophe Soule ^{[3]} tried in his 2004 paper Les varietes sur le corp a un element ^{[4]} to take advantage of these field extensions in order to develop a first approach to an algebraic geometry over . His main guiding idea is stated right at the begining of the paper:

On part de l’idÃ©e qu’une varietÃ© (de type fini) sur doit avoir une extension des scalaires Ã , qui sera un schÃ©ma de type fini sur …

Which means

We start from the idea that a variety (of finite type) over must admit an extension of scalars to , which will be an scheme of finite type over …

Since one of our mantras is that we should avoid hitting directly on the elusive , trying to work over collections of objects defined over it, the most appropriate reformulation of the notion of scheme ^{[5]} that we may use is the, shortest ever, definition (of course equivalent to the classical one):

An scheme over a ground field (or ring) is a covariant functor from the category of -algebras to the category of sets which is locally representable ^{[6]} by an algebra.

With this definition, if we have an scheme over , given by the functor , and a -algebra, the extension of scalars satisfy that for every -algebra we have an inclusion satisfying certain universal property. If we want to do the same thing with in the place of and in the place of , 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 (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 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 should have rank over the integers, and that hints that the following equation should be true:

In particular, for any hypothetical variety over we must have

Going a bit further, we shall assume that the extension of scalars from to induces an equivalence of categories between some (yet to be defined) category of algebras over and the category of rings flat and of finite type over , 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 should be. But that will come in the next post!

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URL to article: **http://cage.ugent.be/~kthas/Fun/index.php/the-skeleton-of-soules-f_un-geometry.html**

URLs in this post:

[1] Image: **http://cage.ugent.be/~kthas/Fun/index.php/the-f_un-folklore.html**

[2] absolute linear algebra: **http://cage.ugent.be/~kthas/Fun/index.php/f_un-linear-algebra.html**

[3] Christophe Soule: **http://www.ihes.fr/~soule/**

[4] Les varietes sur le corp a un element: **http://cage.ugent.be/~kthas/Fun/index.php/soule2004.html**

[5] scheme: **http://en.wikipedia.org/wiki/Scheme_(mathematics)**

[6] representable: **http://en.wikipedia.org/wiki/Representable_functor**

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