\(p\)-adic periods and \(p\)-adic étale cohomology.

*(English)*Zbl 0632.14016
Current trends in arithmetical algebraic geometry, Proc. Summer Res. Conf., Arcata/Calif. 1985, Contemp. Math. 67, 179-207 (1987).

[For the entire collection see Zbl 0615.00004.]

This paper deals with the problem of determining the relation between the de Rham cohomology of a smooth and proper variety over a p-adic field and the p-adic étale cohomology of the extension to an algebraically closure of the field. The de Rham cohomology has the supplementary structure of its Hodge filtration and - at least when the variety has good reduction - the structure of an F-isocrystal, that is it is the extension to the base field of a vector space over the maximal absolutely unramified subfield which has a semilinear automorphism. The p-adic étale cohomology admits a representation of the Galois group of the base field. In general there are functors between objects possessing these two kinds of structures and the conjecture is that the two different cohomology objects correspond under these functors and that they also belong to certain subcategories on which these two functors are equivalences. A positive solution to this conjecture would have consequences for instance to the structure of the p-adic Galois- representation on étale cohomology - such as it being Hodge-Tate.

The article discusses some other such consequences and then goes on to prove the conjecture for varieties defined over an absolutely unramified p-adic field having good reduction and of dimension less than p. The essential technical step is the introduction of the class of maps over the variety which are étale on the generic fibre and are everywhere flat local complete intersection maps. This class is on the one hand sufficiently general to be able to catch the étale cohomology of the generic fibre, on the other hand sufficiently special to have agreeable properties - it is for example reasonably well behaved with respect to crystalline cohomology.

This paper deals with the problem of determining the relation between the de Rham cohomology of a smooth and proper variety over a p-adic field and the p-adic étale cohomology of the extension to an algebraically closure of the field. The de Rham cohomology has the supplementary structure of its Hodge filtration and - at least when the variety has good reduction - the structure of an F-isocrystal, that is it is the extension to the base field of a vector space over the maximal absolutely unramified subfield which has a semilinear automorphism. The p-adic étale cohomology admits a representation of the Galois group of the base field. In general there are functors between objects possessing these two kinds of structures and the conjecture is that the two different cohomology objects correspond under these functors and that they also belong to certain subcategories on which these two functors are equivalences. A positive solution to this conjecture would have consequences for instance to the structure of the p-adic Galois- representation on étale cohomology - such as it being Hodge-Tate.

The article discusses some other such consequences and then goes on to prove the conjecture for varieties defined over an absolutely unramified p-adic field having good reduction and of dimension less than p. The essential technical step is the introduction of the class of maps over the variety which are étale on the generic fibre and are everywhere flat local complete intersection maps. This class is on the one hand sufficiently general to be able to catch the étale cohomology of the generic fibre, on the other hand sufficiently special to have agreeable properties - it is for example reasonably well behaved with respect to crystalline cohomology.

Reviewer: T.Ekedahl

##### MSC:

14F30 | \(p\)-adic cohomology, crystalline cohomology |

14G20 | Local ground fields in algebraic geometry |

14F40 | de Rham cohomology and algebraic geometry |

11S15 | Ramification and extension theory |