Abstract
Let G be a finite group and F a field. We give an elementary proof that the group of normalized cohomological invariants of G over F with values in the Brauer group is isomorphic to \({{\,\mathrm{\mathrm {H}}\,}}^{2}(G,F^{\times })\).
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References
Auslander, M., Goldman, O.: The Brauer group of a commutative ring. Trans. Am. Math. Soc. 97, 367–409 (1960)
Bailey, V.: Cohomological invariants of finite groups. University of California Los Angeles, PhD-thesis (2017)
Blinstein, S., Merkurjev, A.: Cohomological invariants of algebraic tori. Algebra Number Theory 7, 1643–1684 (2013)
Bogomolov, F.: The Brauer group of quotient spaces of linear representations. Izv. Akad. Nauk SSSR Ser. Mat. 51, 485–516, 688 (1987); translation in Math. USSR-Izv. 30, 455–485 (1988)
Bourbaki, N.: Éléments de mathématique. Algèbre commutative. Chapitre 8. Dimension. Chapitre 9. Anneaux locaux noethériens complets, Masson, Paris, (1983)
DeMeyer, F., Ingraham, E.: Separable algebras over commutative rings. Springer Lect. Notes Math. 181, Springer-Verlag, Berlin-Heidelberg-New York (1971)
Draxl, P.: Skew fields, London Mathematical Society Lecture Note Series, 81. Cambridge University Press, Cambridge (1983)
Ford, T.: Separable algebras, Graduate Studies in Mathematics, 183. American Mathematical Society, Providence, RI (2017)
Garibaldi, S., Merkurjev, A., Serre, J.-P.: Cohomological invariants in Galois cohomology, University Lecture Series, 28. American Mathematical Society, Providence, RI (2003)
Gille, P., Szamuely, T.: Central simple algebras and Galois cohomology, Cambridge Studies in Advanced Mathematics, 101. Cambridge University Press, Cambridge (2006)
Hoechsmann, K.: Zum Einbettungsproblem. J. Reine Angew. Math. 229, 81–106 (1968)
Huppert, B.: Endliche Gruppen I, Die Grundlehren der Mathematischen Wissenschaften, 134. Springer, Berlin-New York (1967)
Huppert, B.: Character theory of finite groups, De Gruyter Expositions in Mathematics, 25. Walter de Gruyter & Co., Berlin (1998)
Jacobson, N.: Finite-dimensional division algebras over fields. Springer, Berlin (1996)
Knus, M., Merkurjev, A., Rost, M., Tignol, J.-P. : The book of involutions. With a preface in French by J. Tits, American Mathematical Society Colloquium Publications, 44, American Mathematical Society, Providence, RI, (1998)
Riehm, C.: Extension of scalars in the corestriction of an algebra. J. Pure Appl. Algebra 41, 79–86 (1986)
Serre, J.-P.: Local fields, Translated from the French by Marvin Jay Greenberg, Graduate Texts in Mathematics, 67. Springer, New York-Berlin (1979)
Totaro, B.: Cohomological invariants in positive characteristic, Int. Math. Res. Not. (to appear)
Witt, E.: Schiefkörper über diskret bewerteten Körpern. J. Reine Angew. Math. 176, 153–156 (1937)
Acknowledgements
I would like to thank David McNeilly for discussions about and around the Schur multiplier of a finite group, and Volodya Chernousov for useful comments.
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The author has been supported by an NSERC Grant.
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Gille, S. On Brauer invariants of finite groups. Math. Z. 300, 217–234 (2022). https://doi.org/10.1007/s00209-021-02773-z
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DOI: https://doi.org/10.1007/s00209-021-02773-z