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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

Two-primary algebraic $K$-theory
of rings of integers in number fields


Authors: J. Rognes, C. Weibel and appendix by M. Kolster
Journal: J. Amer. Math. Soc. 13 (2000), 1-54
MSC (2000): Primary 19D50; Secondary 11R70, 11S70, 14F20, 19F27
Published electronically: August 23, 1999
MathSciNet review: 1697095
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Abstract | References | Similar Articles | Additional Information

Abstract: We relate the algebraic $K$-theory of the ring of integers in a number field $F$ to its étale cohomology. We also relate it to the zeta-function of $F$ when $F$ is totally real and Abelian. This establishes the $2$-primary part of the ``Lichtenbaum conjectures.'' To do this we compute the $2$-primary $K$-groups of $F$ and of its ring of integers, using recent results of Voevodsky and the Bloch-Lichtenbaum spectral sequence, modified for finite coefficients in an appendix. A second appendix, by M. Kolster, explains the connection to the zeta-function and Iwasawa theory.


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Additional Information

J. Rognes
Affiliation: Department of Mathematics, University of Oslo, Oslo, Norway
Email: rognes@math.uio.no

C. Weibel
Affiliation: Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903-2101
Email: weibel@math.rutgers.edu

appendix by M. Kolster
Affiliation: Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1
Email: kolster@mcmail.CIS.McMaster.CA

DOI: http://dx.doi.org/10.1090/S0894-0347-99-00317-3
PII: S 0894-0347(99)00317-3
Keywords: Two-primary algebraic $K$-theory, number fields, Lichtenbaum--Quillen conjectures, étale cohomology, motivic cohomology, Bloch--Lichtenbaum spectral sequence
Received by editor(s): July 13, 1998
Published electronically: August 23, 1999
Article copyright: © Copyright 1999 American Mathematical Society