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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.
  • [2] M. Artin and J. L. Verdier, Seminar on étale cohomology of number fields, Lecture Notes of Summer Institute on Algebraic Geometry, Woods Hole, 1964.
  • [3] M. Bienenfeld, Values of zeta- and $ L$-functions at zero: The case of a non-totally imaginary algebraic number field, Ph.D. dissertation, Cornell Univ., 1982.
  • [4] Stephen Lichtenbaum, Values of zeta and 𝐿-functions at zero, Journées Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, Bordeaux, 1974) Soc. Math. France, Paris, 1975, pp. 133–138. Astérisque, Nos. 24–25. MR 0401711 (53 #5538)
  • [5] S. Lichtenbaum and M. Bienenfeld, Values of zeta- and $ L$-functions at zero: a cohomological characterization (to appear).
  • [6] Barry Mazur, Notes on étale cohomology of number fields, Ann. Sci. École Norm. Sup. (4) 6 (1973), 521–552 (1974). MR 0344254 (49 #8993)
  • [7] J.-P. Serre, Cohomologie galoisienne, Lecture Notes in Math., vol. 5, Springer-Verlag, 1964.
  • [8] J. Tate, The cohomology groups of algebraic number fields, Proc. Internat. Congress Math., Amsterdam, 1954.
  • [9] T. Zink, Étale cohomology and duality in number fields, Haberland, Galois cohomology, Berlin, 1978, Appendix 2.

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Additional Information

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

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