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Crystalline Cohomology of Algebraic Stacks and Hyodo-Kato Cohomology
Martin C. Olsson, University of California, Berkeley, CA
A publication of the Société Mathématique de France.
2007; 412 pp; softcover
Number: 316
ISBN-10: 2-85629-249-6
ISBN-13: 978-2-85629-249-5
List Price: US$132
Individual Members: US$118.80
Order Code: AST/316
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In this text the author uses stack-theoretic 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 Rham-Witt complexes for algebraic stacks and applies it to the construction and study of the \((\varphi , N, G)\)-structure on de Rham cohomology. Using the stack-theoretic 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.


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 Rham-Witt theory
  • The abstract Hyodo-Kato 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
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