Does a universal Frobenius map exist? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T17:35:04Z http://mathoverflow.net/feeds/question/6840 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/6840/does-a-universal-frobenius-map-exist Does a universal Frobenius map exist? Marc Nieper-Wißkirchen 2009-11-25T20:30:12Z 2009-12-15T11:44:43Z <p>For any prime p, one has the Frobenius homomorphism F<sub>p</sub> defined on rings of characteristic p.</p> <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<sub>p</sub> over R as "the" base change of F from U to R?</p> <p>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<sub>p</sub> 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<sub>p</sub> of the identity functor on the category of F<sub>p</sub>-algebras. In this naive picture one cannot expect the existence of an f such that f<sub>p</sub> is the Frobenius on F<sub>p</sub>-algebras for all primes p. But is there way to make this picture work?</p> <p>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<sub>p</sub> of A' in case R' is of prime characteristic.</p> <p>(Feel free to modify my two pictures to make them work.)</p> http://mathoverflow.net/questions/6840/does-a-universal-frobenius-map-exist/7324#7324 Answer by Mikhail Bondarko for Does a universal Frobenius map exist? Mikhail Bondarko 2009-12-01T00:36:55Z 2009-12-01T00:36:55Z <p>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 <a href="http://arxiv.org/abs/0909.0069" rel="nofollow">http://arxiv.org/abs/0909.0069</a> suggested at <a href="http://mathoverflow.net/questions/2300/what-is-the-field-with-one-element" rel="nofollow">http://mathoverflow.net/questions/2300/what-is-the-field-with-one-element</a> could also help.</p> http://mathoverflow.net/questions/6840/does-a-universal-frobenius-map-exist/8334#8334 Answer by javier for Does a universal Frobenius map exist? javier 2009-12-09T09:49:02Z 2009-12-09T09:49:02Z <p>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.</p> <p>You might want to have a look at Borger and Wieland work on <a href="http://wwwmaths.anu.edu.au/~borger/papers/03/paper03.html" rel="nofollow">pleythistic algebras</a>. The "lifting of all Frobeniuses at the same time" gives you an structure of \$\Lambda\$-ring, so the paper <a href="http://wwwmaths.anu.edu.au/~borger/papers/05/paper05.html" rel="nofollow">The basic geometry of Witt vectors</a> by Borger might also be useful.</p>