## On the Witt vector Frobenius

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- by Christopher Davis and Kiran S. Kedlaya PDF
- Proc. Amer. Math. Soc.
**142**(2014), 2211-2226

## 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.## References

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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
- MR Author ID: 349028
- ORCID: 0000-0001-8700-8758
- Email: kedlaya@ucsd.edu
- 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
- © Copyright 2014 Christopher Davis and Kiran S. Kedlaya
- Journal: Proc. Amer. Math. Soc.
**142**(2014), 2211-2226 - MSC (2010): Primary 13F35
- DOI: https://doi.org/10.1090/S0002-9939-2014-11953-8
- MathSciNet review: 3195748