Abstract
The notion of a p-adic superspace is introduced and used to give a transparent construction of the Frobenius map on p-adic cohomology of a smooth projective variety over \({\mathbb{Z}_p}\) (the ring of p-adic integers), as well as an alternative construction of the crystalline cohomology of a smooth projective variety over \({\mathbb{F}_p}\) (finite field with p elements).
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Acknowledgments
We are deeply indebted to V. Vologodsky for his help with understanding of the standard approach to the construction of the Frobenius map and to M.Kontsevich, N.Mazzari and A. Ogus for interesting discussions. The second author would also like to thank the Max Planck Institute for Mathematics in Bonn where part of the work on this paper was completed.
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Communicated by A. Connes
Partly supported by NSF grant No. DMS 0505735.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Schwarz, A., Shapiro, I. p-adic Superspaces and Frobenius. Commun. Math. Phys. 282, 87–113 (2008). https://doi.org/10.1007/s00220-008-0526-1
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DOI: https://doi.org/10.1007/s00220-008-0526-1