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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

   

 

On the Witt vector Frobenius


Authors: Christopher Davis and Kiran S. Kedlaya
Journal: Proc. Amer. Math. Soc. 142 (2014), 2211-2226
MSC (2010): Primary 13F35
Published electronically: March 20, 2014
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Abstract: We study the kernel and cokernel of the Frobenius map on the $ p$-typical Witt vectors of a commutative ring, not necessarily of characteristic $ p$. We give many equivalent conditions to surjectivity of the Frobenus map on both finite and infinite length Witt vectors. In particular, surjectivity on finite Witt vectors turns out to be stable under certain integral extensions; this provides a clean formulation of a strong generalization of Faltings's almost purity theorem from $ p$-adic Hodge theory, incorporating recent improvements by Kedlaya and Liu, and by Scholze.


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Additional Information

Christopher Davis
Affiliation: Department of Mathematics, University of California Irvine, Irvine, California 92697
Address at time of publication: Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 København Ø, Denmark
Email: davis@math.ku.dk

Kiran S. Kedlaya
Affiliation: Department of Mathematics, University of California, San Diego, La Jolla, California 92093
Email: kedlaya@ucsd.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2014-11953-8
PII: S 0002-9939(2014)11953-8
Received by editor(s): March 21, 2012
Received by editor(s) in revised form: July 7, 2012
Published electronically: March 20, 2014
Additional Notes: The first author was supported by the Max-Planck-Institut für Mathematik (Bonn).
The second author was supported by the NSF (CAREER grant DMS-0545904, grant DMS-1101343), DARPA (grant HR0011-09-1-0048), MIT (NEC Fund, Green professorship), and UCSD (Warschawski professorship).
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2014 Christopher Davis and Kiran S. Kedlaya