This book presents in great detail a stack-theory approach to p-adic Hodge theory, with special emphasis on a construction of a (ϕ, N, G)-structure on the de Rham cohomology of a proper smooth variety over a p-adic field. The author develops a theory of crystalline cohomology, the de Rham-Witt complex and the Hoydo-Kato isomorphism for certain algebraic stacks. He also reformulates the Kato-Tsuji approach to the semi-stable conjecture (Cst) in this context and gives a general dictionary between his approach and the “usual” log-geometrical method. As an application of the general theory, he deduces new results on the tameness of the Galois action in certain cases of not necessarily semi-stable reduction.

Reviewer:

jnek