I've heard of this many times, but I don't know anything about it.


What I do know is that it is supposed to solve the problem of the fact that the final object in the category of schemes is one-dimensional, namely $\mathop{\text{Spec}}\mathbb Z$.


So, what is the field with one element? And, what are typical geometric objects that descend to $\mathbb F\_1$?



What is the field with one element?
As other have mentioned, F\_1 does not exist of a field. Tits conjectured the existence of a "field of characteristic one" F\_1 for which one would have the equality G(F\_1) = W, where G is any Chevalley group scheme and W its corresponding Weyl group.


Later on Manin suggested that the "absolute point" proposed in Deninger's program to prove the Riemann Hypothesis might be thought of as "Spec F\_1", thus stating the problem of developing an algebraic geometry (and eventually a theory of motives) over it.


There are several non-equivalent approaches to F\_1 geometry, but a common punchline might be "doing F\_1 geometry is finding out the least possible amount of information about an object that still allows to speak about its geometrical properties". A "folkloric" introduction can be found in the paper by Cohn [Projective geometry over F\_1 and the Gaussian binomial coefficients](https://arxiv.org/abs/math/0407093).


It seems that all approaches so far contain a common intersection, consisting on toric varieties which are equivalent to schemes modeled after monoids. In the case of a toric variety, the "descent data" that gives you the F\_1 geometry is the fan structure, that can be reinterpreted as a diagram of monoids (cf. some works by Kato). What else are F\_1 varieties beyond toric is something that depends a lot on your approach, ranging from Kato-Deitmar (for which toric is all there is) to Durov and Haran's categorical constructions which contains very large families of examples. A somehow in-the-middle viewpoint is Soule's (and its refinement by Connes-Consani) which in the finite type case is not restricted only to toric varieties but to something slightly more general (varieties that can be chopped in pieces that are split tori). No approach is yet conclusive, so the definitions and families of examples are likely to change as the theory develops.


Last month Oliver Lorscheid and myself presented an state-of-the-art overview of most of the different approaches to F\_1 geometry: [Mapping F\_1-land: An overview of geometries over the field with one element](https://arxiv.org/abs/0909.0069) (sorry for the self promotion).


