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

Geisser, Thomas; Hesselholt, Lars

doi:10.1090/S0894-0347-05-00505-9

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

Authors: Thomas Geisser and Lars Hesselholt
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
Published electronically: September 16, 2005
MathSciNet review: 2169041
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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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Additional 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

Keywords: de Rham-Witt complex, $p$-adic cohomology, algebraic $K$-theory
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
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.