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Period-index bounds for arithmetic threefolds

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Abstract

The standard period-index conjecture for Brauer groups of p-adic surfaces S predicts that \({{\,\mathrm{ind}\,}}(\alpha )|{{\,\mathrm{per}\,}}(\alpha )^3\) for every \(\alpha \in {{\,\mathrm{Br}\,}}(\mathbf {Q}_p(S))\). Using Gabber’s theory of prime-to-\(\ell \) alterations and the deformation theory of twisted sheaves, we prove that \({{\,\mathrm{ind}\,}}(\alpha )|{{\,\mathrm{per}\,}}(\alpha )^4\) for \(\alpha \) of period prime to 6p, giving the first uniform period-index bounds over such fields.

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References

  1. Abramovich, D., Karu, K.: Weak semistable reduction in characteristic 0. Invent. Math. 139(2), 241–273 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Antieau, B., Williams, B.: Unramified division algebras do not always contain Azumaya maximal orders. Invent. Math. 197(1), 47–56 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. Artin, M.: Brauer–Severi varieties. In: Brauer groups in ring theory and algebraic geometry (Wilrijk, 1981), Lecture Notes in Math., vol. 917, pp. 194–210. Springer, Berlin (1982)

  4. Auel, A., Brussel, E., Garibaldi, S., Vishne, U.: Open problems in central simple algebras. Transform. Groups 16(1), 219–264 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Bosch, S., Lütkebohmert, W., Raynaud, M.: Néron Models. Springer, Berlin (1990)

    Book  MATH  Google Scholar 

  7. Bourbaki, N.: Éléments de mathématique. Algèbre commutative. Chapitres 1 à 4. Masson, Paris (1985)

  8. Cesnavicius, K.: Purity for the Brauer group. accepted for publication at Duke Math. J. (2017)

  9. Colliot-Thélène, J.L.: Birational invariants, purity and the Gersten conjecture. In: \(K\)-theory and algebraic geometry: connections with quadratic forms and division algebras (Santa Barbara, CA, 1992), Proc. Sympos. Pure Math., vol. 58, pp. 1–64. Am. Math. Soc., Providence, RI (1995)

  10. Colliot-Thélène, J.L.: Algèbres simples centrales sur les corps de fonctions de deux variables (d’après A. J. de Jong). Astérisque (307), Exp. No. 949, ix, 379–413 (2006). Séminaire Bourbaki. Vol. 2004/2005

  11. Colliot-Thélène, J.L.: Groupe de Chow des zéro-cycles sur les variétés \(p\)-adiques (d’après S. Saito, K. Sato et al.). Astérisque (339), Exp. No. 1012, vii, 1–30 (2011). Séminaire Bourbaki. Vol. 2009/2010

  12. Conrad, B.: Math 776. From normal crossings to strict normal crossings. http://math.stanford.edu/~conrad/252Page/handouts/crossings.pdf. Accessed July 4, 2018

  13. Cox, D.A., Little, J.B., Schenck, H.K.: Toric Varieties, Graduate Studies in Mathematics, vol. 124. American Mathematical Society, Providence, RI (2011)

    Google Scholar 

  14. de Jong, A.J.: Smoothness, semi-stability and alterations. Inst. Hautes Études Sci. Publ. Math. 83, 51–93 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  15. de Jong, A.J.: A result of Gabber (2003). https://www.math.columbia.edu/~dejong/papers/2-gabber.pdf. Accessed 23 Jan 2019

  16. de Jong, A.J.: The period-index problem for the Brauer group of an algebraic surface. Duke Math. J. 123(1), 71–94 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Eisenbud, D.: Commutative Algebra, with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, vol. 150. Springer, New York (1995)

    MATH  Google Scholar 

  18. Faltings, G., Chai, C.L.: Degeneration of abelian varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 22. Springer, Berlin (1990). With an appendix by David Mumford

  19. Gabber, O.: Some theorems on Azumaya algebras. In: The Brauer group (Sem., Les Plans-sur-Bex, 1980), Lecture Notes in Math., vol. 844, pp. 129–209. Springer, Berlin (1981)

  20. Gille, P., Szamuely, T.: Central Simple Algebras and Galois Cohomology, Cambridge Studies in Advanced Mathematics, vol. 101. Cambridge University Press, Cambridge (2006)

    Book  MATH  Google Scholar 

  21. Grothendieck, A.: Éléments de géométrie algébrique. II. étude globale élémentaire de quelques classes de morphismes. Inst. Hautes Études Sci. Publ. Math. (8), 222 (1961)

  22. Grothendieck, A.: Éléments de géométrie algébrique. III. étude cohomologique des faisceaux cohérents. I. Inst. Hautes Études Sci. Publ. Math. (11), 167 (1961)

  23. Grothendieck, A.: Le groupe de Brauer. II. Théorie cohomologique. In: Dix Exposés sur la Cohomologie des Schémas, pp. 67–87. North-Holland, Amsterdam (1968)

  24. Grothendieck, A.: Le groupe de Brauer. III. Exemples et compléments. In: Dix Exposés sur la Cohomologie des Schémas, pp. 88–188. North-Holland, Amsterdam (1968)

  25. Harbater, D., Hartmann, J., Krashen, D.: Applications of patching to quadratic forms and central simple algebras. Invent. Math. 178(2), 231–263 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Hoobler, R.T.: A cohomological interpretation of Brauer groups of rings. Pacific J. Math. 86(1), 89–92 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  27. Illusie, L.: Complexe cotangent et déformations. I, Lecture Notes in Mathematics, vol. 239. Springer, Berlin (1971)

  28. Illusie, L.: On Gabber’s refined uniformization (2009). Lectures at the University of Tokyo

  29. Illusie, L., Laszlo, Y., Orgogozo, F.: Travaux de Gabber sur l’uniformisation locale et la cohomologie étale des schémas quasi-excellents. Séminaire à l’École polytechnique 2006–2008, Astérisque pp. 363–364 (2014)

  30. Kato, K.: A Hasse principle for two-dimensional global fields. J. Reine Angew. Math. 366, 142–183 (1986). With an appendix by Jean-Louis Colliot-Thélène

  31. Kempf, G., Knudsen, F.F., Mumford, D., Saint-Donat, B.: Toroidal Embeddings I Lecture Notes in Mathematics, vol. 339. Springer, Berlin (1973)

    Book  MATH  Google Scholar 

  32. Knudsen, F.F., Mumford, D.: The projectivity of the moduli space of stable curves. I. Preliminaries on “det” and “Div”. Math. Scand. 39(1), 19–55 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  33. Kozlov, D.: Combinatorial Algebraic Topology, Algorithms and Computation in Mathematics, vol. 21. Springer, Berlin (2008)

    Google Scholar 

  34. Leep, D.B.: The \(u\)-invariant of \(p\)-adic function fields. J. Reine Angew. Math. 679, 65–73 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  35. Lieblich, M.: Moduli of twisted sheaves. Duke Math. J. 138(1), 23–118 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  36. Lieblich, M.: Twisted sheaves and the period-index problem. Compos. Math. 144(1), 1–31 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  37. Lieblich, M.: Period and index in the Brauer group of an arithmetic surface. J. Reine Angew. Math. 659, 1–41 (2011). With an appendix by Daniel Krashen

  38. Lieblich, M.: The period-index problem for fields of transcendence degree 2. Ann. Math. 182(2), 391–427 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  39. Matzri, E.: Symbol length in the Brauer group of a field. Trans. Am. Math. Soc. 368(1), 413–427 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  40. Milnor, J.: Introduction to algebraic \(K\)-theory. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo (1971). Annals of Mathematics Studies, No. 72

  41. Parimala, R., Suresh, V.: Period-index and \(u\)-invariant questions for function fields over complete discretely valued fields. Invent. Math. 197(1), 215–235 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  42. Payne, S.: Boundary complexes and weight filtrations. Mich. Math. J 62(2), 293–322 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  43. Pirutka, A.: A bound to kill the ramification over function fields. J. Algebra 377, 173–178 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  44. Saltman, D.J.: Division algebras over \(p\)-adic curves. J. Ramanujan Math. Soc. 12, 25–47 (1997). Correction, J. Ramanujan Math. Soc. 13 (1998), no. 2, 125–129

  45. Saltman, D.J.: Cyclic algebras over \(p\)-adic curves. J. Algebra 314(2), 817–843 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  46. Thomason, R.W., Trobaugh, T.: Higher algebraic \(K\)-theory of schemes and of derived categories. In: The Grothendieck Festschrift, Vol. III, Progr. Math., vol. 88, pp. 247–435. Birkhäuser, Boston, MA (1990)

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Acknowledgements

We would like to thank the Structured Quartet Research Ensembles (SQuaREs) program of the American Institute of Mathematics (AIM) for its hospitality and support for this project. We also thank the Banff International Research Station (BIRS) and the Institute for Computational and Experimental Research in Mathematics (ICERM) for wonderful working environments during the final stages of preparation of this paper. We thank François Charles, Jean-Louis Colliot-Thélène, Ronen Mukamel, R. Parimala, Eryn Schultz, Lenny Taelman, and Tony Várilly-Alvarado for helpful discussions. We especially thank Dan Abramovich, Sam Payne, and Dhruv Ranganathan for expert advice on toroidal geometry. Finally, we are very grateful to Minseon Shin for several comments and corrections on an earlier draft and to the referees for their detailed comments on this paper.

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Authors and Affiliations

  1. Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, Illinois, USA

    Benjamin Antieau

  2. Department of Mathematics, Yale University, New Haven, Connecticut, USA

    Asher Auel

  3. School of Mathematics and Statistics, Carlton University, Ottawa, Ontario, Canada

    Colin Ingalls

  4. Department of Mathematics, Rutgers, The State University of New Jersey, Newark, New Jersey, USA

    Daniel Krashen

  5. Department of Mathematics, University of Washington, Seattle, Washington, USA

    Max Lieblich

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  1. Benjamin Antieau
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Correspondence to Benjamin Antieau.

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Ben Antieau was partially supported by NSF CAREER Grant DMS-1552766 and NSF Grants DMS-1358832 and DMS-1461847. Asher Auel was partially supported by NSA Young Investigator Grants H98230-13-1-0291 and H98230-16-1-0321. Colin Ingalls was partially supported by an NSERC Discovery grant. Daniel Krashen was partially supported by NSF CAREER Grant DMS-1151252 and FRG Grant DMS-1463901. Max Lieblich was partially supported by NSF CAREER Grant DMS-1056129, NSF Grant DMS-1600813, and a Simons Foundation Fellowship.

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Antieau, B., Auel, A., Ingalls, C. et al. Period-index bounds for arithmetic threefolds. Invent. math. 216, 301–335 (2019). https://doi.org/10.1007/s00222-019-00860-x

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