Azumaya algebras and obstructions to quadratic pairs over a scheme
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- by Philippe Gille, Erhard Neher and Cameron Ruether;
- Trans. Amer. Math. Soc.
- DOI: https://doi.org/10.1090/tran/9472
- Published electronically: June 10, 2025
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Abstract:
We investigate quadratic pairs for Azumaya algebras with involutions over a base scheme $S$ as defined by Calmès and Fasel, generalizing the case of quadratic pairs on central simple algebras over a field (Knus, Merkurjev, Rost, Tignol). We describe a cohomological obstruction for an Azumaya algebra over $S$ with orthogonal involution to admit a quadratic pair and provide a classification of all quadratic pairs it admits. When $S$ is affine this obstruction vanishes, however it is non-trivial in general. In particular, we construct explicit examples with non-trivial obstructions.References
- G. de Rham, Lectures on introduction to algebraic topology, Tata Institute of Fundamental Research Lectures on Mathematics, No. 44, Tata Institute of Fundamental Research, Bombay, 1969. Notes by V. J. Lal. MR 370553
- N. Bourbaki, Éléments de mathématique. Algèbre. Chapitre 8. Modules et anneaux semi-simples, Springer, Berlin, 2012 (French). Second revised edition of the 1958 edition [MR0098114]. MR 3027127, DOI 10.1007/978-3-540-35316-4
- Michel Brion, Homogeneous bundles over abelian varieties, J. Ramanujan Math. Soc. 27 (2012), no. 1, 91–118. MR 2933488
- Michel Brion, Homogeneous projective bundles over abelian varieties, Algebra Number Theory 7 (2013), no. 10, 2475–2510. MR 3194649, DOI 10.2140/ant.2013.7.2475
- Michel Brion, Homogeneous vector bundles over abelian varieties via representation theory, Represent. Theory 24 (2020), 85–114. MR 4058941, DOI 10.1090/ert/537
- Baptiste Calmès and Jean Fasel, Groupes classiques, Autours des schémas en groupes. Vol. II, Panor. Synthèses, vol. 46, Soc. Math. France, Paris, 2015, pp. 1–133 (French, with English and French summaries). MR 3525594
- A. Grothendieck and J. A. Dieudonné, Éléments de géométrie algébrique. I, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 166, Springer-Verlag, Berlin, 1971 (French). MR 3075000
- Timothy J. Ford, Separable algebras, Graduate Studies in Mathematics, vol. 183, American Mathematical Society, Providence, RI, 2017. MR 3618889, DOI 10.1090/gsm/183
- Jean Giraud, Cohomologie non abélienne, Die Grundlehren der mathematischen Wissenschaften, Band 179, Springer-Verlag, Berlin-New York, 1971 (French). MR 344253
- P. Gille, E. Neher, and C. Ruether, The norm functor over schemes, Preprint, arXiv:2401.15051.
- Philippe Gille and Arturo Pianzola, Galois cohomology and forms of algebras over Laurent polynomial rings, Math. Ann. 338 (2007), no. 2, 497–543. MR 2302073, DOI 10.1007/s00208-007-0086-2
- A. Grothendieck, Le Groupe de Brauer I, Sém. Bourbaki, No. 290, 1964/65.
- U. Görtz and T. Wedhorn, Algebraic Geometry I, second edition, Springer Fachmedien Wiesbaden, 2020.
- Manfred Knebusch, Symmetric bilinear forms over algebraic varieties, Conference on Quadratic Forms—1976 (Proc. Conf., Queen’s Univ., Kingston, Ont., 1976) Queen’s Papers in Pure and Appl. Math., No. 46, Queen’s Univ., Kingston, ON, 1977, pp. 103–283. MR 498378
- Max-Albert Knus, Quadratic and Hermitian forms over rings, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 294, Springer-Verlag, Berlin, 1991. With a foreword by I. Bertuccioni. MR 1096299, DOI 10.1007/978-3-642-75401-2
- Nicholas M. Katz and Barry Mazur, Arithmetic moduli of elliptic curves, Annals of Mathematics Studies, vol. 108, Princeton University Press, Princeton, NJ, 1985. MR 772569, DOI 10.1515/9781400881710
- Max-Albert Knus, Alexander Merkurjev, Markus Rost, and Jean-Pierre Tignol, The book of involutions, American Mathematical Society Colloquium Publications, vol. 44, American Mathematical Society, Providence, RI, 1998. With a preface in French by J. Tits. MR 1632779, DOI 10.1090/coll/044
- Max-Albert Knus and Manuel Ojanguren, Théorie de la descente et algèbres d’Azumaya, Lecture Notes in Mathematics, Vol. 389, Springer-Verlag, Berlin-New York, 1974 (French). MR 417149
- Masaki Kashiwara and Pierre Schapira, Categories and sheaves, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 332, Springer-Verlag, Berlin, 2006. MR 2182076, DOI 10.1007/3-540-27950-4
- Qing Liu, Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, vol. 6, Oxford University Press, Oxford, 2002. Translated from the French by Reinie Erné; Oxford Science Publications. MR 1917232
- James S. Milne, Étale cohomology, Princeton Mathematical Series, No. 33, Princeton University Press, Princeton, NJ, 1980. MR 559531
- Shigeru Mukai, Semi-homogeneous vector bundles on an Abelian variety, J. Math. Kyoto Univ. 18 (1978), no. 2, 239–272. MR 498572, DOI 10.1215/kjm/1250522574
- Zinovy Reichstein and Boris Youssin, Essential dimensions of algebraic groups and a resolution theorem for $G$-varieties, Canad. J. Math. 52 (2000), no. 5, 1018–1056. With an appendix by János Kollár and Endre Szabó. MR 1782331, DOI 10.4153/CJM-2000-043-5
- Alex Rosenberg and Daniel Zelinsky, Automorphisms of separable algebras, Pacific J. Math. 11 (1961), 1109–1117. MR 148709
- Chih Han Sah, Symmetric bilinear forms and quadratic forms, J. Algebra 20 (1972), 144–160. MR 294378, DOI 10.1016/0021-8693(72)90094-4
- Jean-Pierre Serre, Sur la topologie des variétés algébriques en caractéristique $p$, Symposium internacional de topología algebraica International symposium on algebraic topology, Universidad Nacional Autónoma de México and UNESCO, México, 1958, pp. 24–53 (French). MR 98097
- Schémas en groupes. III: Structure des schémas en groupes réductifs, Lecture Notes in Mathematics, Vol. 153, Springer-Verlag, Berlin-New York, 1970 (French). Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3); Dirigé par M. Demazure et A. Grothendieck. MR 274460
- The Stacks Project Authors, Stacks project, http://stacks.math.columbia.edu/.
- J. Tits, Formes quadratiques, groupes orthogonaux et algèbres de Clifford, Invent. Math. 5 (1968), 19–41 (French). MR 230747, DOI 10.1007/BF01404536
- William C. Waterhouse, A unified Kummer-Artin-Schreier sequence, Math. Ann. 277 (1987), no. 3, 447–451. MR 891585, DOI 10.1007/BF01458325
Bibliographic Information
- Philippe Gille
- Affiliation: UMR 5208 du CNRS, Institut Camille Jordan, Université Claude Bernard Lyon 1, 43 boulevard du 11 novembre 1918, 69622 Villeurbanne cedex, France
- MR Author ID: 333189
- ORCID: 0000-0001-8066-5835
- Email: gille@math.univ-lyon1.fr
- Erhard Neher
- Affiliation: Department of Mathematics and Statistics, University of Ottawa, 150 Louis-Pasteur Private, Ottawa, Ontario K1N 9A7, Canada
- MR Author ID: 190247
- ORCID: 0000-0001-7783-8539
- Email: Erhard.Neher@uottawa.ca
- Cameron Ruether
- Affiliation: The “Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest, Romania
- MR Author ID: 1417255
- Email: cameronruether@gmail.com
- Received by editor(s): July 13, 2023
- Received by editor(s) in revised form: November 14, 2023, and January 27, 2025
- Published electronically: June 10, 2025
- Additional Notes: The first author was supported by the Labex Milyon (ANR-10-LABX-0070) of Université de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX- 0007) operated by the French National Research Agency (ANR). The research of the second author was partially supported by an NSERC grant. The research of the third author was partially supported by the NSERC grants of the second author and of Kirill Zainoulline at the University of Ottawa (2022) and partially supported by the NSERC grants of Mikhail Kotchetov and Yorck Sommerhäuser at the Memorial University of Newfoundland (2023).
- © Copyright 2025 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
- MSC (2020): Primary 11E81; Secondary 11E39, 14F20, 16H05, 20G10, 20G35
- DOI: https://doi.org/10.1090/tran/9472