Galois groups and cohomological functors

Efrat, Ido; Mináč, Ján

doi:10.1090/tran/6724

Galois groups and cohomological functors
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by Ido Efrat and Ján Mináč PDF

Trans. Amer. Math. Soc. 369 (2017), 2697-2720 Request permission

Abstract:

Let $q=p^s$ be a prime power, $F$ a field containing a root of unity of order $q$, and $G_F$ its absolute Galois group. We determine a new canonical quotient $\mathrm {Gal}(F_{(3)}/F)$ of $G_F$ which encodes the full mod-$q$ cohomology ring $H^*(G_F,\mathbb {Z}/q)$ and is minimal with respect to this property. We prove some fundamental structure theorems related to these quotients. In particular, it is shown that when $q=p$ is an odd prime, $F_{(3)}$ is the compositum of all Galois extensions $E$ of $F$ such that $\mathrm {Gal}(E/F)$ is isomorphic to $\{1\}$, $\mathbb {Z}/p$ or to the nonabelian group $H_{p^3}$ of order $p^3$ and exponent $p$.

References

Eberhard Becker, Euklidische Körper und euklidische Hüllen von Körpern, J. Reine Angew. Math. 268(269) (1974), 41–52 (German). MR 354625, DOI 10.1515/crll.1974.268-269.41
Fedor Bogomolov and Yuri Tschinkel, Introduction to birational anabelian geometry, Current developments in algebraic geometry, Math. Sci. Res. Inst. Publ., vol. 59, Cambridge Univ. Press, Cambridge, 2012, pp. 17–63. MR 2931864
Sunil K. Chebolu, Ido Efrat, and Ján Mináč, Quotients of absolute Galois groups which determine the entire Galois cohomology, Math. Ann. 352 (2012), no. 1, 205–221. MR 2885583, DOI 10.1007/s00208-011-0635-6
Ido Efrat, The Zassenhaus filtration, Massey products, and representations of profinite groups, Adv. Math. 263 (2014), 389–411. MR 3239143, DOI 10.1016/j.aim.2014.07.006
Ido Efrat, Filtrations of free groups as intersections, Arch. Math. (Basel) 103 (2014), no. 5, 411–420. MR 3281289, DOI 10.1007/s00013-014-0701-x
Ido Efrat and Ján Mináč, On the descending central sequence of absolute Galois groups, Amer. J. Math. 133 (2011), no. 6, 1503–1532. MR 2863369, DOI 10.1353/ajm.2011.0041
Ido Efrat and Ján Mináč, Small Galois groups that encode valuations, Acta Arith. 156 (2012), no. 1, 7–17. MR 2997568, DOI 10.4064/aa156-1-2
Michael D. Fried and Moshe Jarden, Field arithmetic, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 11, Springer-Verlag, Berlin, 2005. MR 2102046
Philippe Gille and Tamás Szamuely, Central simple algebras and Galois cohomology, Cambridge Studies in Advanced Mathematics, vol. 101, Cambridge University Press, Cambridge, 2006. MR 2266528, DOI 10.1017/CBO9780511607219
Klaus Hoechsmann, Zum Einbettungsproblem, J. Reine Angew. Math. 229 (1968), 81–106 (German). MR 244190, DOI 10.1515/crll.1968.229.81
Irving Kaplansky, Infinite abelian groups, Revised edition, University of Michigan Press, Ann Arbor, Mich., 1969. MR 0233887
Helmut Koch, Galois theory of $p$-extensions, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2002. With a foreword by I. R. Shafarevich; Translated from the 1970 German original by Franz Lemmermeyer; With a postscript by the author and Lemmermeyer. MR 1930372, DOI 10.1007/978-3-662-04967-9
John P. Labute, Demuškin groups of rank $\aleph _{0}$, Bull. Soc. Math. France 94 (1966), 211–244. MR 222177
A. S. Merkur′ev and A. A. Suslin, $K$-cohomology of Severi-Brauer varieties and the norm residue homomorphism, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 5, 1011–1046, 1135–1136 (Russian). MR 675529
Ján Mináč and Michel Spira, Formally real fields, Pythagorean fields, $C$-fields and $W$-groups, Math. Z. 205 (1990), no. 4, 519–530. MR 1082872, DOI 10.1007/BF02571260
Ján Mináč and Michel Spira, Witt rings and Galois groups, Ann. of Math. (2) 144 (1996), no. 1, 35–60. MR 1405942, DOI 10.2307/2118582
Ján Mináč, John Swallow, and Adam Topaz, Galois module structure of $(\ell ^n)$th classes of fields, Bull. Lond. Math. Soc. 46 (2014), no. 1, 143–154. MR 3161770, DOI 10.1112/blms/bdt082
Ján Mináč and Nguyễn Duy Tân, The kernel unipotent conjecture and the vanishing of Massey products for odd rigid fields, Adv. Math. 273 (2015), 242–270. MR 3311763, DOI 10.1016/j.aim.2014.12.028
Ján Mináč and Nguyen Duy Tân, Triple Massey products and Galois theory, J. Eur. Math. Soc., to appear.
Jürgen Neukirch, Alexander Schmidt, and Kay Wingberg, Cohomology of number fields, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 323, Springer-Verlag, Berlin, 2008. MR 2392026, DOI 10.1007/978-3-540-37889-1
Florian Pop, On the birational anabelian program initiated by Bogomolov I, Invent. Math. 187 (2012), no. 3, 511–533. MR 2891876, DOI 10.1007/s00222-011-0331-x
Luis Ribes and Pavel Zalesskii, Profinite groups, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 40, Springer-Verlag, Berlin, 2010. MR 2599132, DOI 10.1007/978-3-642-01642-4
Jean-Pierre Serre, Sur la dimension cohomologique des groupes profinis, Topology 3 (1965), 413–420 (French). MR 180619, DOI 10.1016/0040-9383(65)90006-6
Jean-Pierre Serre, Galois cohomology, Corrected reprint of the 1997 English edition, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2002. Translated from the French by Patrick Ion and revised by the author. MR 1867431
Adam Topaz, Commuting-liftable subgroups of Galois groups II, J. reine angew. Math. (2012), to appear, available at arXiv:1208.0583v5.
Fernando Rodriguez Villegas, Relations between quadratic forms and certain Galois extensions, a manuscript, Ohio State University, 1988, http://www.math.utexas.edu/users/villegas/osu.pdf.
Vladimir Voevodsky, Motivic cohomology with $\textbf {Z}/2$-coefficients, Publ. Math. Inst. Hautes Études Sci. 98 (2003), 59–104. MR 2031199, DOI 10.1007/s10240-003-0010-6
Vladimir Voevodsky, On motivic cohomology with $\mathbf Z/l$-coefficients, Ann. of Math. (2) 174 (2011), no. 1, 401–438. MR 2811603, DOI 10.4007/annals.2011.174.1.11
Charles Weibel, 2007 Trieste lectures on the proof of the Bloch-Kato conjecture, Some recent developments in algebraic $K$-theory, ICTP Lect. Notes, vol. 23, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2008, pp. 277–305. MR 2509183
C. Weibel, The norm residue isomorphism theorem, J. Topol. 2 (2009), no. 2, 346–372. MR 2529300, DOI 10.1112/jtopol/jtp013