## The de Rham-Witt complex and $p$-adic vanishing cycles

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- by Thomas Geisser and Lars Hesselholt
- J. Amer. Math. Soc.
**19**(2006), 1-36 - DOI: https://doi.org/10.1090/S0894-0347-05-00505-9
- Published electronically: September 16, 2005
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## Abstract:

We determine the structure of the reduction modulo $p$ of the absolute de Rham-Witt complex of a smooth scheme over a discrete valuation ring of mixed characteristic $(0,p)$ with log-poles along the special fiber and show that the sub-sheaf fixed by the Frobenius map is isomorphic to the sheaf of $p$-adic vanishing cycles. We use this result together with the main results of*op. cit.*to evaluate the algebraic $K$-theory with finite coefficients of the quotient field of the henselian local ring at a generic point of the special fiber. The result affirms the Lichtenbaum-Quillen conjecture for this field.

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## Bibliographic Information

**Thomas Geisser**- Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90089
- Email: geisser@math.usc.edu
**Lars Hesselholt**- Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 and Department of Mathematics, Nagoya University, Nagoya, Japan
- MR Author ID: 329414
- Email: larsh@math.mit.edu; larsh@math.nagoya-u.ac.jp
- Received by editor(s): January 5, 2004
- Published electronically: September 16, 2005
- Additional Notes: A previous version of this paper was entitled
*On the $K$-theory of a henselian discrete valuation field with non-perfect residue field*.

The authors were supported in part by grants from the National Science Foundation. The first author received additional support from the Japan Society for the Promotion of Science and the Alfred P. Sloan Foundation - © Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc.
**19**(2006), 1-36 - MSC (2000): Primary 11G25, 11S70; Secondary 14F30, 19D55
- DOI: https://doi.org/10.1090/S0894-0347-05-00505-9
- MathSciNet review: 2169041