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Strong linkage for function fields of surfaces

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Abstract

Over a global field any finite number of central simple algebras of exponent dividing m is split by a common cyclic field extension of degree m. We show that the same property holds for function fields of 2-dimensional excellent schemes over a henselian local domain of dimension one or two with algebraically closed residue field.

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References

  1. Auslander, M., Brumer, A.: Brauer group of discrete valuation rings. Nederl. Akad. Wetensch. Proc. Ser. A 71, 286–296 (1968)

    Article  MathSciNet  Google Scholar 

  2. Auslander, M., Goldman, O.: The Brauer group of a commutative ring. Trans. Am. Math. Soc. 97, 367–409 (1960)

    Article  MathSciNet  Google Scholar 

  3. Ax, J.: Proof of some conjectures on cohomological dimension. Proc. Am. Math. Soc. 16, 1214–1221 (1965)

    Article  MathSciNet  Google Scholar 

  4. Chapman, A., Tignol, J.-P.: Linkage of Pfister forms over \({\mathbb{C}}(x_1,\ldots , x_n)\). Ann. K-Theory 4, 521–524 (2019)

    Article  MathSciNet  Google Scholar 

  5. Colliot-Thélène, J..-L., Ojanguren, M., Parimala, R.: Quadratic forms over fraction fields of \(2\)-dimensional henselian rings and Brauer groups of related schemes. Tata Inst. Fund. Res. Mumbai Part II 16, 185–217 (2002)

    MathSciNet  MATH  Google Scholar 

  6. Česnavičius, K.: Purity for the Brauer group. Duke Math. J. 168, 1461–1486 (2019)

    Article  MathSciNet  Google Scholar 

  7. de Jong, A.J.: The period-index problem for the Brauer group of an algebraic surface. Duke Math. J. 123, 71–94 (2004)

    MathSciNet  MATH  Google Scholar 

  8. Dieudonné, J., Grothendieck, A.: Éléments de géométrie algébrique: IV. Inst. Hautes Études Sci. Publ. Math. 32, 5–361 (1967)

    Article  Google Scholar 

  9. Draxl, P.: Skew Fields. London Mathematical Society Lecture Note Series, vol. 81. Cambridge University Press, Cambridge (1983)

    Book  Google Scholar 

  10. Efrat, I.: A Hasse principle for function fields over PAC fields. Israel J. Math. 122, 43–60 (2001)

    Article  MathSciNet  Google Scholar 

  11. Engler, A.J., Prestel, A.: Valued Fields. Springer Monographs in Mathematics, Springer-Verlag, Berlin (2005)

    MATH  Google Scholar 

  12. Ford, T.J.: Brauer group of a curve over a strictly local discrete valuation ring. Israel J. Math. 96, 259–266 (1996)

    Article  MathSciNet  Google Scholar 

  13. Ford, T.J., Saltman, D.J.: Division algebras over Henselian surfaces. Ring theory 1989 (Ramat Gan and Jerusalem, 1988/1989). Israel Mathematical Conference Proceedings, vol. 1, pp. 320–336

  14. Fried, M.D., Jarden, M.: Field arithmetic. Third edition. Revised by Jarden. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vol. 11. Springer-Verlag, Berlin (2008)

  15. Gille, P., Szamuely, T.: Central Simple Algebras and Galois Cohomology. Cambridge University Press, Cambridge (2006)

    Book  Google Scholar 

  16. Grothendieck, A.: Éléments de géométrie algébrique: II. Étude globale élémentaire de quelques classes de morphismes. Publications Mathématiques de l’I.H.E.S 8, 5–222 (1961)

    Article  Google Scholar 

  17. Grothendieck, A.: Le groupe de Brauer, \(iii\), Exemples et compléments. In: Dix exposés sur la cohomologie des schemas. Advanced Studies in Pure Mathematics, vol. 3, pp. 88–188. North-Holland, Amsterdam (1968)

  18. Lenstra, H. W.: \({\sf K}_2\) of global fields consists of symbols. Algebraic \(K\)-theory (Proceedings Conference, Northwestern University, Evanston, IL, 1976). Lecture Notes in Mathematics, vol. 551, pp. 69–73. Springer, Berlin (1976)

  19. Lipman, J.: Introduction to resolution of singularities. Algebraic geometry (Proceedings of Symposium on Pure Mathematics, Humboldt State University, Arcata, CA, 1974), vol. 29, pp. 187–230. American Mathematical Society, Providence, RI (1975)

  20. Lipman, J.: Desingularization of two-dimensional schemes. Ann. Math. 107, 151–207 (1978)

    Article  MathSciNet  Google Scholar 

  21. Liu, Q.: Algebraic Geometry and Arithmetic Curves. Oxford Graduate Texts in Mathematics, Oxford University Press, Oxford (2012)

    Google Scholar 

  22. Matsumura, H.: Commutative Ring Theory. Cambridge Studies in Advanced Mathematics, vol. 8. Cambridge University Press, Cambridge (1986)

    MATH  Google Scholar 

  23. Milnor, J.: Algebraic \(K\)-theory and quadratic forms. Invent. Math. 9, 318–344 (1970)

    Article  MathSciNet  Google Scholar 

  24. Parimala, R., Suresh, V.: Isotropy of quadratic forms over function fields of \(p\)-adic curves. Inst. Hautes Études Sci. Publ. Math. No. 88, 129–150 (1998)

    Article  MathSciNet  Google Scholar 

  25. Parimala, R., Suresh, V.: The \(u\)-invariant of the function fields of \(p\)-adic curves. Ann. Math. 172, 1391–1405 (2010)

    Article  MathSciNet  Google Scholar 

  26. Pfister, A.: Quadratic Forms with Applications to Algebraic Geometry and Topology. London Mathematical Society Lecture Notes Series, vol. 217. Cambridge University Press, Cambridge (1995)

    Book  Google Scholar 

  27. Reiner, I.: Maximal Orders. London Mathematical Society Monographs New Series, Oxford University Press, Oxford (2003)

    MATH  Google Scholar 

  28. Saltman, D.J.: Division algebras over \(p\)-adic curves. J. Raman. Math. Soc. 12, 25–47 (1997)

    MathSciNet  MATH  Google Scholar 

  29. Saltman, D.J.: Lectures on Division Algberas. CBMS Regional Conference Series in Mathematics, vol. 94. American Mathematical Society, Providence (1999)

    Book  Google Scholar 

  30. Saltman, D.J.: Cyclic algebras over \(p\)-adic curves. J. Algebra 314, 817–843 (2007)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors express their gratitude to Jean-Louis Colliot-Thélène, Arno Fehm, David Grimm, Gonzalo Manzano Flores, R. Parimala, Suresh Venapally and Jan Van Geel for various answers, discussions, suggestions, simplifications and other valuable input related to this article. They further gratefully acknowledge the referee’s comments and suggestions, which helped to streamline the presentation.

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Authors and Affiliations

  1. Departement Wiskunde, Universiteit Antwerpen, Middelheimlaan 1, 2020, Antwerpen, Belgium

    Karim Johannes Becher & Parul Gupta

  2. Institut für Algebra, Technische Universität Dresden, 01062, Dresden, Germany

    Parul Gupta

  3. IISER Pune, Dr. Homi Bhabha Road, Pashan, Pune, 411 008, India

    Parul Gupta

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  1. Karim Johannes Becher
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Correspondence to Karim Johannes Becher.

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This work was supported by the Fonds Wetenschappelijk Onderzoek – Vlaanderen in the FWO Odysseus Programme (project Explicit Methods in Quadratic Form Theory) and by the Bijzonder Onderzoeksfonds, University of Antwerp (project BOF-DOCPRO-4, 2865)

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Becher, K.J., Gupta, P. Strong linkage for function fields of surfaces. manuscripta math. 168, 181–201 (2022). https://doi.org/10.1007/s00229-021-01301-x

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