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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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An étale cohomology duality theorem for number fields with a real embedding
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by Mel Bienenfeld PDF
Trans. Amer. Math. Soc. 303 (1987), 71-96 Request permission

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
    M. Artin, Grothendieck topologies, mimeographed notes, Harvard Univ., 1962. M. Artin and J. L. Verdier, Seminar on étale cohomology of number fields, Lecture Notes of Summer Institute on Algebraic Geometry, Woods Hole, 1964. 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.
  • Stephen Lichtenbaum, Values of zeta and $L$-functions at zero, Journées Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, Bordeaux, 1974) Astérisque, Nos. 24–25, Soc. Math. France, Paris, 1975, pp. 133–138. MR 0401711
    • S. Lichtenbaum and M. Bienenfeld, Values of zeta- and $L$-functions at zero: a cohomological characterization (to appear).
  • Barry Mazur, Notes on étale cohomology of number fields, Ann. Sci. École Norm. Sup. (4) 6 (1973), 521–552 (1974). MR 344254, DOI 10.24033/asens.1257
    • J.-P. Serre, Cohomologie galoisienne, Lecture Notes in Math., vol. 5, Springer-Verlag, 1964. J. Tate, The cohomology groups of algebraic number fields, Proc. Internat. Congress Math., Amsterdam, 1954. T. Zink, Étale cohomology and duality in number fields, Haberland, Galois cohomology, Berlin, 1978, Appendix 2.
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Additional Information
  • © Copyright 1987 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 303 (1987), 71-96
  • MSC: Primary 12G05; Secondary 11R34, 11R42, 14F20
  • DOI: https://doi.org/10.1090/S0002-9947-1987-0896008-0
  • MathSciNet review: 896008