Astérisque 2007; 412 pp; softcover Number: 316 ISBN10: 2856292496 ISBN13: 9782856292495 List Price: US$132 Individual Members: US$118.80 Order Code: AST/316
 In this text the author uses stacktheoretic techniques to study the crystalline structure on the de Rham cohomology of a proper smooth scheme over a \(p\)adic field and applications to \(p\)adic Hodge theory. He develops a general theory of crystalline cohomology and de RhamWitt complexes for algebraic stacks and applies it to the construction and study of the \((\varphi , N, G)\)structure on de Rham cohomology. Using the stacktheoretic point of view instead of log geometry, he develops the ingredients needed to prove the \(C_{\text {st}}\)conjecture using the method of Fontaine, Messing, Hyodo, Kato, and Tsuji, except for the key computation of \(p\)adic vanishing cycles. He also generalizes the construction of the monodromy operator to schemes with more general types of reduction than semistable and proves new results about tameness of the action of Galois on cohomology. A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list. Readership Graduate students and research mathematicians interested in algebra and algebraic geometry. Table of Contents  Introduction
 Divided power structures on stacks and the crystalline topos
 Crystals and differential calculus on stacks
 The Cartier isomorphism and applications
 De RhamWitt theory
 The abstract HyodoKato isomorphism
 The \((\varphi, N, G)\)structure on de Rham cohomology
 A variant construction of the \((\varphi, N, G)\) structure
 Comparison with syntomic cohomology
 Comparison with log geometry in the sense of Fontaine and Illusie
 Bibliography
 Index of notation
 Index of terminology
