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Bulletin of the London Mathematical Society Advance Access originally published online on April 29, 2008
Bulletin of the London Mathematical Society 2008 40(3):439-446; doi:10.1112/blms/bdn024
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© 2008 London Mathematical Society

Galois theory and integral models of {Lambda}-rings

James Borger

Mathematical Sciences Institute
Building 27
Australian National University
Canberra ACT 0200
Australia

Bart de Smit

Mathematisch Instituut
Universiteit Leiden
Postbus 9512
2300 RA Leiden
The Netherlands
desmit@math.leidenuniv.nl

Received 31 May 2007. Revision received 21 October 2007.

We show that any {Lambda}-ring, in the sense of Riemann–Roch theory, which is finite étale over the rational numbers and has an integral model as a {Lambda}-ring is contained in a product of cyclotomic fields. In fact, we show that the category of such {Lambda}-rings is described in a Galois-theoretic way in terms of the monoid of pro-finite integers under multiplication and the cyclotomic character. We also study the maximality of these integral models and give a more precise, integral version of the result above. These results reveal an interesting relation between {Lambda}-rings and the class field theory.

2000 Mathematics Subject Classification 13K05 (primary), 11R37, 19L20, 16W99 (secondary).


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