Integral crystalline cohomology over very ramified valuation rings

Abstract

We explain how to set up an integral version ( Z p \mathbb {Z}_{p} as opposed to Q p \mathbb {Q}_{p} ) of Fontaine’s comparison between crystalline and étale cohomology, over p p -adic fields with arbitrary ramification index. The main results then are that Fontaine’s map respects integrality of Tate-cycles, and a construction of versal deformations of p p -divisible groups with Tate-cycles. An appendix deals with finite generation of crystalline cohomology.