A motivic conjecture of Milne

Details A motivic conjecture of Milne A motivic conjecture of Milne

2003-08-20

arxiv.org/abs/math/0308202v5

Mathematics

Number Theory

60 pages. Final version to appear in J. Reine Agew. Math. (Crelle)

Scientific paper

Let k be an algebraically closed field of characteristic p>0. Let W(k) be the ring of Witt vectors with coefficients in k. We prove a motivic conjecture of Milne that relates, in the case of abelian schemes, the \'etale cohomology with $\dbZ_p$ coefficients to the crystalline cohomology with integral coefficients, in the more general context of p-divisible groups endowed with {\it arbitrary} families of crystalline tensors over a finite, discrete valuation ring extension of W(k). This extends a result of Faltings in [Fa2]. As a main new tool we construct global deformations of p-divisible groups endowed with crystalline tensors over certain regular, formally smooth schemes over W(k) whose special fibers over k have a Zariski dense set of k-valued points.

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