Crystalline cohomology of algebraic stacks and Hyodo–Kato cohomology

Martin C. Olsson

In this text we study using stack–theoretic techniques the crystalline structure on the de Rham cohomology of a complete smooth variety over a p–adic field. Such a structure is known to exists by the CdR–conjecture of Fontaine now proven independently by Faltings, Niziol, and Tsuji, and is intimately tied to the action of Galois on the p–adic ´etale cohomology.

The paper contains two main parts. In the first part, we develop a general theory of crystalline cohomology for algebraic stacks. We study in the stack theoretic setting generalizations of the basic definitions and results of crystalline cohomology, the correspondence between crystals and modules with integrable connection, the Cartier isomorphism and Cartier descent, Ogus’ generalization of Mazur’s Theorem, as well as a stack–theoretic generalization of the de Rham–Witt complex.

The second part is devoted to applying these ideas and techniques to the construction and study of the so–called (‘,N,G)–structure on the de Rham cohomology of a smooth proper variety over a p–adic field. Using the stack–theoretic point of view instead of log geometry, we develop the ingredients needed to prove the Cst–conjecture using the method of Fontaine, Messing, Hyodo, Kato, and Tsuji, except for the key computation of p–adic vanishing cycles. In addition we give a new construction of the so–called Hyodo–Kato isomorphism and its generalizations needed for the proof of the Cst–conjecture based on a classification of F–crystals “up to almost isomorphism” over a certain ring Whti. Using the stack–theoretic approach we also generalize the construction of the monodromy operator to schemes with more general types of reduction than semi–stable, and prove new results about tameness of the action of Galois on the (‘,N,G)–module arising from schemes with so–called “log smooth reduction”. Finally there is a chapter explaining the relationship between the stack–theoretic approach and the approach using logarithmic geometry.