Relative Brauer groups
of discrete valued fields
Authors:
Burton Fein and Murray Schacher
Journal:
Proc. Amer. Math. Soc. 127 (1999), 677-684
MSC (1991):
Primary 12G05; Secondary 12E15
DOI:
https://doi.org/10.1090/S0002-9939-99-04792-9
MathSciNet review:
1487365
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: Let be a non-trivial finite Galois extension of a field
. In this paper we investigate the role that valuation-theoretic properties of
play in determining the non-triviality of the relative Brauer group,
, of
over
. In particular, we show that when
is finitely generated of transcendence degree 1 over a
-adic field
and
is a prime dividing
, then the following conditions are equivalent: (i) the
-primary component,
, is non-trivial, (ii)
is infinite, and (iii) there exists a valuation
of
trivial on
such that
divides the order of the decomposition group of
at
.
- [B] Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York, 1994. Corrected reprint of the 1982 original. MR 1324339
- [FKS] Burton Fein, William M. Kantor, and Murray Schacher, Relative Brauer groups. II, J. Reine Angew. Math. 328 (1981), 39–57. MR 636194, https://doi.org/10.1515/crll.1981.328.39
- [FSS1] Burton Fein, David J. Saltman, and Murray Schacher, Brauer-Hilbertian fields, Trans. Amer. Math. Soc. 334 (1992), no. 2, 915–928. MR 1075382, https://doi.org/10.1090/S0002-9947-1992-1075382-0
- [FSS2] Burton Fein, David J. Saltman, and Murray Schacher, Crossed products over rational function fields, J. Algebra 156 (1993), no. 2, 454–493. MR 1216479, https://doi.org/10.1006/jabr.1993.1084
- [FS1] Burton Fein and Murray Schacher, A conjecture about relative Brauer groups, 𝐾-theory and algebraic geometry: connections with quadratic forms and division algebras (Santa Barbara, CA, 1992) Proc. Sympos. Pure Math., vol. 58, Amer. Math. Soc., Providence, RI, 1995, pp. 161–169. MR 1327296
- [FS2] Burton Fein and Murray Schacher, Crossed products over algebraic function fields, J. Algebra 171 (1995), no. 2, 531–540. MR 1315911, https://doi.org/10.1006/jabr.1995.1026
- [Hw] Yoon Sung Hwang, The corestriction of valued division algebras over Henselian fields. I, Pacific J. Math. 170 (1995), no. 1, 53–81. MR 1359972
- [JW] Bill Jacob and Adrian Wadsworth, Division algebras over Henselian fields, J. Algebra 128 (1990), no. 1, 126–179. MR 1031915, https://doi.org/10.1016/0021-8693(90)90047-R
- [La] S. Lang, Algebraic Number Theory, Addison Wesley, Reading, 1968.
- [Li] Stephen Lichtenbaum, Duality theorems for curves over 𝑝-adic fields, Invent. Math. 7 (1969), 120–136. MR 242831, https://doi.org/10.1007/BF01389795
- [M] B. Heinrich Matzat, Konstruktive Galoistheorie, Lecture Notes in Mathematics, vol. 1284, Springer-Verlag, Berlin, 1987 (German). MR 1004467
- [Pi] Richard S. Pierce, Associative algebras, Graduate Texts in Mathematics, vol. 88, Springer-Verlag, New York-Berlin, 1982. Studies in the History of Modern Science, 9. MR 674652
- [Po] Florian Pop, Galoissche Kennzeichnung 𝑝-adisch abgeschlossener Körper, J. Reine Angew. Math. 392 (1988), 145–175 (German). MR 965062, https://doi.org/10.1515/crll.1988.392.145
- [R] Peter Roquette, Analytic theory of elliptic functions over local fields, Hamburger Mathematische Einzelschriften (N.F.), Heft 1, Vandenhoeck & Ruprecht, Göttingen, 1970. MR 0260753
- [Sa] Shuji Saito, Class field theory for curves over local fields, J. Number Theory 21 (1985), no. 1, 44–80. MR 804915, https://doi.org/10.1016/0022-314X(85)90011-3
- [Se] Jean-Pierre Serre, Corps locaux, Publications de l’Institut de Mathématique de l’Université de Nancago, VIII, Actualités Sci. Indust., No. 1296. Hermann, Paris, 1962 (French). MR 0150130
- [T] J. T. Tate, Global class field theory, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., 1967, pp. 162–203. MR 0220697
- [W] Edwin Weiss, Algebraic number theory, McGraw-Hill Book Co., Inc., New York-San Francisco-Toronto-London, 1963. MR 0159805
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Additional Information
Burton Fein
Affiliation:
Department of Mathematics, Oregon State University, Corvallis, Oregon 97331
Email:
fein@math.orst.edu
Murray Schacher
Affiliation:
Department of Mathematics, University of California at Los Angeles, Los Angeles, California 90024
Email:
mms@math.ucla.edu
DOI:
https://doi.org/10.1090/S0002-9939-99-04792-9
Keywords:
Brauer group,
discrete valued field
Received by editor(s):
June 23, 1997
Additional Notes:
The authors are grateful for support under NSA Grants MDA904-95-H-1054 and MDA904-95-H-1022, respectively.
Communicated by:
Ken Goodearl
Article copyright:
© Copyright 1999
American Mathematical Society