Raskind, Wayne A finiteness theorem in the Galois cohomology of algebraic number fields. (English) Zbl 0648.12009 Trans. Am. Math. Soc. 303, 743-749 (1987). Soit \(k\) un corps de nombres, \(\mathfrak g= \mathrm{Gal}(\bar k/k)\) et \(M\) un \(\mathbb{Z}_{\ell}\)-module de type fini muni d’une action continue de \(\mathfrak g\). Si \(M\) est non ramifié presque partout, et si pour les places non ramifiées non au-dessus de \(\ell\), l’élément de Frobenius n’a de points of higher order and certain sequences of orderings of higher level called “chains”. In the paper under review the author studies these chains and shows that this study can be reduced to tractable problems in abelian group theory and in the theory of real closures of level 1, i.e., the real closed fields introduced by Artin and Schreier. Moreover, the author’s methods enable him to study chains under field extensions, subfield formation, and places. There is a paper by N. Schwartz dealing with related questions [J. Algebra 110, 74–107 (1987; Zbl 0632.12022)] which, it should be noted, was submitted over three years after the paper under review was. Reviewer: Alex Rosenberg (Santa Barbara) Cited in 5 Documents MSC: 11R34 Galois cohomology 14K15 Arithmetic ground fields for abelian varieties 14G25 Global ground fields in algebraic geometry 14F20 Étale and other Grothendieck topologies and (co)homologies Keywords:étale cohomology; Mordell-Weil theorem; algebraic cycles Citations:Zbl 0632.12022 PDFBibTeX XMLCite \textit{W. Raskind}, Trans. Am. Math. Soc. 303, 743--749 (1987; Zbl 0648.12009) Full Text: DOI References: [1] Spencer Bloch and Arthur Ogus, Gersten’s conjecture and the homology of schemes, Ann. Sci. École Norm. Sup. (4) 7 (1974), 181 – 201 (1975). · Zbl 0307.14008 [2] S. Bloch, Torsion algebraic cycles and a theorem of Roitman, Compositio Math. 39 (1979), no. 1, 107 – 127. · Zbl 0463.14002 [3] Spencer Bloch, Algebraic cycles and values of \?-functions, J. Reine Angew. Math. 350 (1984), 94 – 108. · Zbl 0527.14008 · doi:10.1515/crll.1984.350.94 [4] Jean-Louis Colliot-Thélène, Jean-Jacques Sansuc, and Christophe Soulé, Torsion dans le groupe de Chow de codimension deux, Duke Math. J. 50 (1983), no. 3, 763 – 801 (French). · Zbl 0574.14004 · doi:10.1215/S0012-7094-83-05038-X [5] Jean-Louis Colliot-Thélène and Wayne Raskind, \?\(_{2}\)-cohomology and the second Chow group, Math. Ann. 270 (1985), no. 2, 165 – 199. · Zbl 0536.14004 · doi:10.1007/BF01456181 [6] Serge Lang, Fundamentals of Diophantine geometry, Springer-Verlag, New York, 1983. · Zbl 0528.14013 [7] Wayne Raskind, ”Le théorème de Mordell-Weil faible” pour \?\(^{0}\)(\?,\?\(_{2}\))/\?\(_{2}\)\?, C. R. Acad. Sci. Paris Sér. I Math. 299 (1984), no. 7, 241 – 244 (French, with English summary). · Zbl 0597.14009 [8] Peter Schneider, Über gewisse Galoiscohomologiegruppen, Math. Z. 168 (1979), no. 2, 181 – 205 (German). · Zbl 0421.12024 · doi:10.1007/BF01214195 [9] -, Thesis, Regensburg, 1980. [10] J.-P. Serre, Cohomologie galoisienne, Lecture Notes in Math., vol. 5, Springer-Verlag, Berlin and New York, 1964. · Zbl 0143.05901 [11] Christophe Soulé, On higher \?-adic regulators, Algebraic \?-theory, Evanston 1980 (Proc. Conf., Northwestern Univ., Evanston, Ill., 1980) Lecture Notes in Math., vol. 854, Springer, Berlin-New York, 1981, pp. 372 – 401. [12] A. A. Suslin, Mennicke symbols and their applications in the \?-theory of fields, Algebraic \?-theory, Part I (Oberwolfach, 1980) Lecture Notes in Math., vol. 966, Springer, Berlin-New York, 1982, pp. 334 – 356. · Zbl 0502.18004 [13] John Tate, Relations between \?\(_{2}\) and Galois cohomology, Invent. Math. 36 (1976), 257 – 274. · Zbl 0359.12011 · doi:10.1007/BF01390012 [14] SGA \( 4\tfrac{1} {2}\) par P. Deligne, avec la collaboration de J. F. Boutot, A. Grothendieck, L. Illusie et J.-L. Verdier, Lecture Notes in Math., vol. 569, Springer-Verlag, Berlin and New York, 1977. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.