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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

An étale cohomology duality theorem for number fields with a real embedding


Author: Mel Bienenfeld
Journal: Trans. Amer. Math. Soc. 303 (1987), 71-96
MSC: Primary 12G05; Secondary 11R34, 11R42, 14F20
MathSciNet review: 896008
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Abstract: The restriction on $ 2$-primary components in the Artin-Verdier duality theorem [2] has been eliminated by Zink [9], who has shown that the sheaf of units for the étale topology over the ring of integers of any number field acts as a dualizing sheaf for a modified cohomology of sheaves. The present paper provides an alternate means of removing the $ 2$-primary restriction. Like Zink's, it involves a topology which includes infinite primes, but it avoids modified cohomology and will be more directly applicable in the proof of a theorem of Lichtenbaum regarding zeta- and $ L$-functions [4, 5]. Related results--including the cohomology of units sheaves, the norm theorem, and punctual duality theorem of Mazur [6]--are also affected by the use of a topology including the infinite primes. The corresponding results in the new setting are included here.


References [Enhancements On Off] (What's this?)

  • [1] M. Artin, Grothendieck topologies, mimeographed notes, Harvard Univ., 1962.
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  • [9] T. Zink, Étale cohomology and duality in number fields, Haberland, Galois cohomology, Berlin, 1978, Appendix 2.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1987-0896008-0
PII: S 0002-9947(1987)0896008-0
Keywords: Artin-Verdier duality theorem, étale cohomology, real algebraic number field
Article copyright: © Copyright 1987 American Mathematical Society