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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The Cech filtration and monodromy in log crystalline cohomology

Author: Elmar Grosse-Klönne
Journal: Trans. Amer. Math. Soc. 359 (2007), 2945-2972
MSC (2000): Primary 14F30
Published electronically: January 26, 2007
MathSciNet review: 2286064
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Abstract: For a strictly semistable log scheme $ Y$ over a perfect field $ k$ of characteristic $ p$ we investigate the canonical Cech spectral sequence $ (C)_T$ abutting the Hyodo-Kato (log crystalline) cohomology $ H_{crys}^*(Y/T)_{\mathbb{Q}}$ of $ Y$ and beginning with the log convergent cohomology of its various component intersections $ Y^i$. We compare the filtration on $ H_{crys}^*(Y/T)_{\mathbb{Q}}$ arising from $ (C)_T$ with the monodromy operator $ N$ on $ H_{crys}^*(Y/T)_{\mathbb{Q}}$. We also express $ N$ through residue maps and study relations with singular cohomology. If $ Y$ lifts to a proper strictly semistable (formal) scheme $ X$ over a finite totally ramified extension of $ W(k)$, with generic fibre $ X_K$, we obtain results on how the simplicial structure of $ X_K^{an}$ (as analytic space) is reflected in $ H_{dR}^*(X_K)=H_{dR}^*(X_K^{an})$.

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Elmar Grosse-Klönne
Affiliation: Mathematisches Institut der Universität Münster, Einsteinstrasse 62, 48149 Mün- ster, Germany

Keywords: Logarithmic crystalline cohomology, monodromy operator, weight filtration, Steenbrink complex, analytic spaces
Received by editor(s): January 5, 2005
Received by editor(s) in revised form: July 11, 2005
Published electronically: January 26, 2007
Additional Notes: Most of this work was done during my visit at the University of California, Berkeley. I wish to thank Robert Coleman (and Bishop) for welcoming me there so warmly. Thanks are also due to Ehud de Shalit, Yukiyoshi Nakkajima and Arthur Ogus for useful related discussions. I thank the referee for his careful reading of the manuscript and his suggestions for improving the exposition. I am grateful to the Deutsche Forschungsgemeinschaft for supporting my stay at Berkeley.
Article copyright: © Copyright 2007 American Mathematical Society