The Cech filtration and monodromy in log crystalline cohomology
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- by Elmar Grosse-Klönne PDF
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Abstract:
For a strictly semistable log scheme $Y$ over a perfect field $k$ of characteristic $p$ we investigate the canonical Čech 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 $H_{dR}^*(X_K)=H_{dR}^*(X_K^{an})$.References
- Gil Alon and Ehud de Shalit, Cohomology of discrete groups in harmonic cochains on buildings, Israel J. Math. 135 (2003), 355–380. MR 1997050, DOI 10.1007/BF02776064
- Vladimir G. Berkovich, An analog of Tate’s conjecture over local and finitely generated fields, Internat. Math. Res. Notices 13 (2000), 665–680. MR 1772523, DOI 10.1155/S1073792800000362
- Bruno Chiarellotto, Rigid cohomology and invariant cycles for a semistable log scheme, Duke Math. J. 97 (1999), no. 1, 155–169. MR 1682272, DOI 10.1215/S0012-7094-99-09707-7
- Bruno Chiarellotto and Bernard Le Stum, Sur la pureté de la cohomologie cristalline, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), no. 8, 961–963 (French, with English and French summaries). MR 1649945, DOI 10.1016/S0764-4442(98)80122-5
- Robert Coleman and Adrian Iovita, The Frobenius and monodromy operators for curves and abelian varieties, Duke Math. J. 97 (1999), no. 1, 171–215. MR 1682268, DOI 10.1215/S0012-7094-99-09708-9
- Ehud de Shalit, The $p$-adic monodromy-weight conjecture for $p$-adically uniformized varieties, Compos. Math. 141 (2005), no. 1, 101–120. MR 2099771, DOI 10.1112/S0010437X04000594
- Elmar Grosse-Klönne, Finiteness of de Rham cohomology in rigid analysis, Duke Math. J. 113 (2002), no. 1, 57–91. MR 1905392, DOI 10.1215/S0012-7094-02-11312-X
- Elmar Grosse-Klönne, Compactifications of log morphisms, Tohoku Math. J. (2) 56 (2004), no. 1, 79–104. MR 2028919
- Elmar Grosse-Klönne, Frobenius and monodromy operators in rigid analysis, and Drinfel′d’s symmetric space, J. Algebraic Geom. 14 (2005), no. 3, 391–437. MR 2129006, DOI 10.1090/S1056-3911-05-00402-9
- Osamu Hyodo, On the de Rham-Witt complex attached to a semi-stable family, Compositio Math. 78 (1991), no. 3, 241–260. MR 1106296
- Osamu Hyodo and Kazuya Kato, Semi-stable reduction and crystalline cohomology with logarithmic poles, Astérisque 223 (1994), 221–268. Périodes $p$-adiques (Bures-sur-Yvette, 1988). MR 1293974
- Luc Illusie, Autour du théorème de monodromie locale, Astérisque 223 (1994), 9–57 (French). Périodes $p$-adiques (Bures-sur-Yvette, 1988). MR 1293970
- Tetsushi Ito, Weight-monodromy conjecture for $p$-adically uniformized varieties, Invent. Math. 159 (2005), no. 3, 607–656. MR 2125735, DOI 10.1007/s00222-004-0395-y
- Fumiharu Kato, Log smooth deformation theory, Tohoku Math. J. (2) 48 (1996), no. 3, 317–354. MR 1404507, DOI 10.2748/tmj/1178225336
- Kazuya Kato, Logarithmic structures of Fontaine-Illusie, Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988) Johns Hopkins Univ. Press, Baltimore, MD, 1989, pp. 191–224. MR 1463703
- Nicholas M. Katz and William Messing, Some consequences of the Riemann hypothesis for varieties over finite fields, Invent. Math. 23 (1974), 73–77. MR 332791, DOI 10.1007/BF01405203
- Reinhardt Kiehl, Theorem A und Theorem B in der nichtarchimedischen Funktionentheorie, Invent. Math. 2 (1967), 256–273 (German). MR 210949, DOI 10.1007/BF01425404
- Bernard Le Stum, La structure de Hyodo-Kato pour les courbes, Rend. Sem. Mat. Univ. Padova 94 (1995), 279–301 (French, with French summary). MR 1370917
- A. Mokrane, La suite spectrale des poids en cohomologie de Hyodo-Kato, Duke Math. J. 72 (1993), no. 2, 301–337 (French). MR 1248675, DOI 10.1215/S0012-7094-93-07211-0
- Y. Nakkajima, $p$-adic weight spectral sequences of log varieties, preprint
- A. Ogus, Logarithmic De Rham cohomology, preprint
- Arthur Ogus, $F$-crystals on schemes with constant log structure, Compositio Math. 97 (1995), no. 1-2, 187–225. Special issue in honour of Frans Oort. MR 1355125
- P. Schneider and U. Stuhler, The cohomology of $p$-adic symmetric spaces, Invent. Math. 105 (1991), no. 1, 47–122. MR 1109620, DOI 10.1007/BF01232257
- Atsushi Shiho, Crystalline fundamental groups. II. Log convergent cohomology and rigid cohomology, J. Math. Sci. Univ. Tokyo 9 (2002), no. 1, 1–163. MR 1889223
Additional Information
- Elmar Grosse-Klönne
- Affiliation: Mathematisches Institut der Universität Münster, Einsteinstrasse 62, 48149 Mün- ster, Germany
- Email: klonne@math.uni-muenster.de
- 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.
- © Copyright 2007 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 359 (2007), 2945-2972
- MSC (2000): Primary 14F30
- DOI: https://doi.org/10.1090/S0002-9947-07-04138-4
- MathSciNet review: 2286064