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Vanishing of Massey Products and Brauer Groups

Published online by Cambridge University Press:  20 November 2018

Ido Efrat  and
Eliyahu Matzri
Ido Efrat
Affiliation:
Department of Mathematics, Ben-Gurion University of the Negev, Be’er-Sheva 84105, Israel e-mail: efrat@math.bgu.ac.ilelimatzri@gmail.com
Eliyahu Matzri
Affiliation:
Department of Mathematics, Ben-Gurion University of the Negev, Be’er-Sheva 84105, Israel e-mail: efrat@math.bgu.ac.ilelimatzri@gmail.com
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Abstract

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Let $p$ be a prime number and $F$ a field containing a root of unity of order $p$. We relate recent results on vanishing of triple Massey products in the $\bmod-p $ Galois cohomology of $F$, due to Hopkins, Wickelgren, Mináč, and Tân, to classical results in the theory of central simple algebras. We prove a stronger form of the vanishing property for global fields.

Keywords

16K5011R3412G0512E30Galois cohomologyBrauer groupstriple Massey products, global fields
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 58 , Issue 4 , 01 December 2015 , pp. 730 - 740
Copyright
Copyright © Canadian Mathematical Society 2015

References

[1] Adrian, A. A., Structure of algebras. American Mathematical Society Colloquium Publications, XXIV, American Mathematical Society, Providence, RI, 1939.Google Scholar
[2] Arason, J. Kr., Cohomologische Invarianten quadratischer Formen. J. Algebra 36(1975), 448–491. http://dx.doi.Org/1 0.101 6/0021-8693(75)90145-3 Google Scholar
[3] Artin, E. and Tate, J., Class field theory. AMS Chelsea Publishing, Providence, RI, 2009.Google Scholar
[4] Draxl, P. K., Skew fields. London Mathematical Society Lecture Notes Series, 81, Cambridge University Press, Cambridge, 1983.Google Scholar
[5] Dwyer, W. G., Homology, Massey products and maps between groups. J. Pure Appl. Algebra 6(1975), 177–190. http://dx.doi.Org/10.101 6/0022-4049(75)90006-7 Google Scholar
[6] Efrat, I., The Zassenhaus filtration, Massey products, and representations of profinite groups. Adv. Math. 263(2014), 389–411. http://dx.doi.Org/10.1016/j.aim.2014.07.006 Google Scholar
[7] Efrat, I. and Minâč, J., On the descending central sequence of absolute Galois groups. Amer. J. Math. 133(2011), no. 6, 1503–1532. http://dx.doi.Org/10.1353/ajm.2011.0041 Google Scholar
[8] Elman, R., Lam, T. Y., Tignol, J.-P. and Wadsworth, A. R., Witt rings and Brauer groups under multiquadratic extensions. I. Amer. J. Math. 105(1983), no. 5,1119-1170. http://dx.doi.Org/1 0.2307/2374336 Google Scholar
[9] Fenn, R. A., Techniques of geometric topology. London Mathematical Society Lecture Notes Series, 57, Cambridge University Press, Cambridge, 1983.Google Scholar
[10] Hopkins, M. J. and Wickelgren, K. G., Splitting varieties for triple Massey products. J. Pure Appl. Algebra 219(2015), no. 5, 1304–1319. http://dx.doi.Org/1 0.101 6/j.jpaa.2O1 4.06.006 Google Scholar
[11] Kraines, D., Massey higher products. Trans. Amer. Math. Soc. 124(1966), 431–449. http://dx.doi.Org/10.1090/S0002-9947-1966-0202136-1 Google Scholar
[12] Lemire, N., Minâč, J. and Swallow, J., Galois module structure of Galois cohomology and partial Euler-Poincaré characteristics. J. Reine Angew. Math. 613(2007), 147–173.Google Scholar
[13] Matzri, E., Z3 x Z^-crossedproducts. J. Algebra, 418(2014), 1–7. http://dx.doi.Org/1 0.101 6/j.jalgebra.2O14.06.035 Google Scholar
[14] May, J. P., Matric Massey products. J. Algebra 12(1969), 533–568. http://dx.doi.Org/10.1016/0021-8693(69)90027-1 Google Scholar
[15] McKinnie, K., Degeneracy and decomposability in abelian crossed products. J. Algebra 328(2011), 443–460. http://dx.doi.Org/10.1016/j.jalgebra.2O10.10.029 Google Scholar
[16] Minâč, J. and Tân, N. D., Triple Massey products and Galois theory. J. Eur. Math. Soc, to appear.arxiv:1307.6624Google Scholar
[17] Minâc, J. and Tân, N. D., Triple Massey products over global fields. arxiv:1403.4586Google Scholar
[18] Minâč, J. and Tân, N. D., The kernel unipotent conjecture and the vanishing of Massey products for odd rigid fields. (with an appendix by I. Efrat, J. Minâc, and N.D. Tân) Adv. Math. 273(2015), 242–270. http://dx.doi.Org/10.1016/j.aim.2O14.12.028 Google Scholar
[19] Rowen, L. H., Cyclic division algebras. Israel J. Math. 41(1982), no. 3, 213–234; Correction: Israel J. Math. 43(1982), no. 3, 277-280. http://dx.doi.Org/10.1007/BF02771722 Google Scholar
[20] Rowen, L. H. , Division algebras of exponent 2 and characteristic 2. J. Algebra 90(1984), no. 1, 71–83. http://dx.doi.Org/10.1016/0021-8693(84)90199-6 Google Scholar
[21] Saltman, D. J., Indecomposable division algebras. Comm. Algebra 7(1979), no. 8, 791–817. http://dx.doi.Org/10.1080/00927877908822376 Google Scholar
[22] Serre, J.-P., Local fields. Graduate Texts in Mathematics, 67, Springer-Verlag, New York-Berlin, 1979.Google Scholar
[23] Tignol, J.-P., Central simple algebras with involution. In: Ring theory (Proc. Antwerp Conf. (NATO Adv. Study Inst), Univ. Antwerp, 1978), Lecture Notes in Pure and Appl. Math., 51, Dekker, New York, 1979, pp. 279–285.Google Scholar
[24] Tignol, J.-P., Corps à involution neutralisés par une extension abélienne élémentaire. In: The Brauer Group (Sem., Les Plans-sur-Bex, 1980), Lecture Notes in Math, 844, Springer, Berlin, 1981, pp. 1–34.Google Scholar
[25] Tignol, J.-P., Produits croisés abéliens. J. Algebra 70(1981), no. 2, 420–436. http://dx.doi.Org/10.1016/0021-8693(81)90227-1 Google Scholar
[26] Tignol, J.-P., Algébres indécomposables d'exposant premier. Adv. in Math. 65(1987), 205–228. http://dx.doi.Org/1 0.101 6/0001-8708(87)90022-3 Google Scholar
[27] Voevodsky, V., Motivic cohomology with ï/l-coefficients. Publ. Math. Inst. Hautes Études Sci. 98(2003), 59–104.Google Scholar
[28] Wickelgren, K., n-nilpotent obstructions to U\ sections o/F1 - ﹛0,1, oo﹜ and Massey products. In: Galois-Teichmuller theory and arithmetic geometry, Adv. Stud. Pure Math., 63, Math. Soc. Japan, Tokyo, 2012, pp. 579–600.Google Scholar