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

DOI:
https://doi.org/10.1090/S0002-9947-1987-0896008-0

MathSciNet review:
896008

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Abstract | References | Similar Articles | Additional Information

Abstract: The restriction on -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 -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 -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.

**[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*-*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****[5]**S. Lichtenbaum and M. Bienenfeld,*Values of zeta- and*-*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****[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

DOI:
https://doi.org/10.1090/S0002-9947-1987-0896008-0

Keywords:
Artin-Verdier duality theorem,
étale cohomology,
real algebraic number field

Article copyright:
© Copyright 1987
American Mathematical Society