Hostname: page-component-76fb5796d-wq484 Total loading time: 0 Render date: 2024-04-28T14:38:05.814Z Has data issue: false hasContentIssue false
  • English
  • Français

LOGARITHMIC DE RHAM–WITT COMPLEXES VIA THE DÉCALAGE OPERATOR

Part of: (Co)homology theory Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  26 August 2021

Zijian Yao Open the ORCID record for Zijian Yao [Opens in a new window]
Zijian Yao*
Affiliation:
Department of Mathematics, Harvard University
*
zijian.yao.math@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

We provide a new formalism of de Rham–Witt complexes in the logarithmic setting. This construction generalises a result of Bhatt–Lurie–Mathew and agrees with those of Hyodo–Kato and Matsuue for log-smooth schemes of log-Cartier type. We then use our construction to study the monodromy action and slopes of Frobenius on log crystalline cohomology.

Keywords

de Rham-Witt complexlog crystalline cohomologymonodromy operatorde Rham cohomologyNygaard filtrationslopes of Frobenius

MSC classification

Primary: 14F30: $p$-adic cohomology, crystalline cohomology
Secondary: 14G20: Local ground fields 14G22: Rigid analytic geometry 14F40: de Rham cohomology
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 22 , Issue 3 , May 2023 , pp. 1319 - 1382
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Berthelot, P. and Ogus, A., Notes on Crystalline Cohomology (Princeton University Press, Princeton, NJ, 1978).Google Scholar
Bhatt, B., Lurie, J. and Mathew, A., Revisiting the de Rham-Witt complex, Asterisque N424.Google Scholar
Bhatt, B., Morrow, M. and Scholze, P., Integral $p$ -adic Hodge theory, Publ. Math. Inst. Hautes Études Sci. 128 (2018).CrossRefGoogle Scholar
de Jong, A. J. et al., Stacks project, https://stacks.math.columbia.edu/.Google Scholar
Deligne, P., Théorie de Hodge: II, Publ. Math. Inst. Hautes Études Sci. 40 (1971).CrossRefGoogle Scholar
Gregory, O. and Langer, A., Overconvergent de Rham-Witt cohomology for semistable varieties, Preprint, 2017. https://arxiv.org/abs/1711.09943.Google Scholar
Hesselholt, L. and Madsen, I., On the K-theory of local fields. Ann. Math. (2003).CrossRefGoogle Scholar
Hesselholt, L. and Madsen, I., On the de Rham–Witt complex in mixed characteristic, Ann. Sci. École Norm. Sup. (2004).CrossRefGoogle Scholar
Hyodo, O. and Kato, K.,. Semi-stable reduction and cystalline cohomology with logarithmic poles, Asterisque 223 (1994).Google Scholar
Illusie, L., Complexe de de Rham-Witt et cohomologie cristalline, Ann. Sci. École Norm. Sup. (1979).CrossRefGoogle Scholar
Illusie, L. and Raynaud, M., Les suites spectrales associées au complexe de de Rham-Witt, Publ. Math. Inst. Hautes Études Sci. 57 (1983).CrossRefGoogle Scholar
Joyal, A., $\delta$ -Anneaux et vecteurs de Witt, C. R. Math. Rep. Acad. Sci. Canada (1985).Google Scholar
Kato, K., Logarithmic structures of Fontaine-Illusie, in Algebraic Analysis, Geometry, and Number Theory (ed., Igusa, J.-I.) (Johns Hopkins University Press, Baltimore, MD). 1989, 191–224.Google Scholar
Langer, A. and Zink, T., de Rham-Witt cohomology for a proper and smooth morphism, J. Inst. Math. Jussieu 3 (2004).CrossRefGoogle Scholar
Matsuue, H., On relative and overconvergent de Rham-Witt cohomology for log schemes, Math. Z. (2017).CrossRefGoogle Scholar
Mazur, B., Frobenius and the Hodge filtration, Bull. Amer. Math. Soc (1972).CrossRefGoogle Scholar
Mazur, B., Frobenius and the Hodge filtration, estimates, Ann. Math. (1973).Google Scholar
Nakkajima, Y., $p$ -Adic weight spectral sequences of log varieties, J. Math. Sci. Univ. Tokyo 12 (2005).Google Scholar
Nygaard, N. O., Slopes of powers of Frobenius on crystalline cohomology, Ann. Sci. École Norm. Sup. (1981).Google Scholar
Yao, Z., The log crystalline specialization of Ainf-cohomology in the semi-stable case, Preprint, 2019.Google Scholar