Does a universal Frobenius map exist? - MathOverflow

Does a universal Frobenius map exist?

2009-11-25T20:30:12Z

For any prime p, one has the Frobenius homomorphism F_p defined on rings of characteristic p.

Is there any kind of object, say U, with a "universal Frobenius map" F such that for any prime p and any ring R of characteristic p we can view the Frobenius F_p over R as "the" base change of F from U to R?

I have the following picture in mind: In some sense it should be possible to view the category of Z-algebras as a sheaf of categories over Spec Z such that the fibre over Spec F_p is just the category of F_p-algebras. A natural transformation f of the identity functor on the category of Z-algebras should restrict to a natural transformation f_p of the identity functor on the category of F_p-algebras. In this naive picture one cannot expect the existence of an f such that f_p is the Frobenius on F_p-algebras for all primes p. But is there way to make this picture work?

Another possible way to answer my question could be the following: Is there a classifying topos of, say, algebras with a Frobenius action? By this I mean the following: Is there a topos E with a fixed ring object R and an algebra A over it and an R-linear endomorphism f of A such that for any other topos E' with similar data R', A' there is a unique morphism of topoi E' -> E that pulls back R, A to R', A' and such that f is pulled back to the Frobenius f_p of A' in case R' is of prime characteristic.

(Feel free to modify my two pictures to make them work.)

Answer by Mikhail Bondarko for Does a universal Frobenius map exist?

2009-12-01T00:36:55Z

It seems that you need 'a field with one element' here. There is a very readable paper by Manin on the subject (probably, this is "Lectures on zeta functions and motives (according to Deninger and Kurokawa)", Astérisque 228 (4): 121–163). Probably the survey http://arxiv.org/abs/0909.0069 suggested at http://mathoverflow.net/questions/2300/what-is-the-field-with-one-element could also help.

Answer by javier for Does a universal Frobenius map exist?

2009-12-09T09:49:02Z

I don't really get the categorical picture of what you are asking, but it feels something very similar to the relation between finite fields of characteristic $p$ and the ring (of characteristic 0) of $p$-typical Witt vectors.

You might want to have a look at Borger and Wieland work on pleythistic algebras. The "lifting of all Frobeniuses at the same time" gives you an structure of $\Lambda$-ring, so the paper The basic geometry of Witt vectors by Borger might also be useful.