The period-index problem for fields of transcendence degree $2$

Abstract

Using geometric methods we prove the standard period-index conjecture for the Brauer group of a field of transcendence degree $2$ over $\mathbf{F}_p$.

  • [MR0354655] Go to document Théorie des Intersections et Théorème de Riemann-Roch, Berthelot, P., Grothendieck, A., and Illusie, L., Eds., New York: Springer-Verlag, 1971, vol. 225.
    @book{MR0354655, mrkey = {0354655},
      EDITOR={Berthelot, P. and Grothendieck, A. and Illusie, L.} , title = {Théorie des Intersections et Théorème de {R}iemann-{R}och},
      series = {Lecture Notes in Math},
      volume = {225},
      note = {S{é}minaire de G{é}om{é}trie Alg{é}brique du Bois-Marie 1966--1967 (SGA 6)} , publisher = {Springer-Verlag},
      year = {1971},
      pages = {xii+700},
      mrclass = {14-06},
      mrnumber = {0354655},
      address = {New York},
      zblnumber = {0218.14001},
      doi = {10.1007/BFb0066283},
      }
  • [starr] Go to document H. Xuhua, A. J. de Jong, and J. M. Starr, "Families of rationally simply connected varieties over surfaces and torsors for semisimple groups," Publ. Math. Inst. Hautes Études Sci., vol. 114, pp. 1-85, 2011.
    @article{starr, mrkey = {2854858},
      author = {Xuhua, He and de Jong, Aise Johan and Starr, Jason Michael},
      title = {Families of rationally simply connected varieties over surfaces and torsors for semisimple groups},
      JOURNAL = {Publ. Math. Inst. Hautes Études Sci.},
      FJOURNAL = {Publications Mathématiques. Institut de Scientifiques},
      volume = {114},
      year = {2011},
      pages = {1--85},
      issn = {0073-8301},
      mrclass = {14D06 (14D23 14M22)},
      mrnumber = {2854858},
      zblnumber = {1285.14053},
      doi = {10.1007/s10240-011-0035-1},
      }
  • [MR0260746] M. Artin, "Algebraization of formal moduli. I," in Global Analysis (Papers in Honor of K. Kodaira), Tokyo: Univ. Tokyo Press, 1969, pp. 21-71.
    @incollection{MR0260746, mrkey = {0260746},
      author = {Artin, M.},
      title = {Algebraization of formal moduli. {I}},
      booktitle = {Global {A}nalysis ({P}apers in {H}onor of {K}. {K}odaira)},
      pages = {21--71},
      publisher = {Univ. Tokyo Press},
      address = {Tokyo},
      year = {1969},
      mrclass = {14.05},
      mrnumber = {0260746},
      mrreviewer = {H. Kurke},
      zblnumber = {0205.50402},
      }
  • [MR657428] M. Artin, "Local structure of maximal orders on surfaces," in Brauer Groups in Ring Theory and Algebraic Geometry, New York: Springer-Verlag, 1982, vol. 917, pp. 146-181.
    @incollection{MR657428, mrkey = {0657428},
      author = {Artin, M.},
      title = {Local structure of maximal orders on surfaces},
      booktitle = {Brauer Groups in Ring Theory and Algebraic Geometry},
      venue = {{W}ilrijk, 1981},
      series = {Lecture Notes in Math.},
      volume = {917},
      pages = {146--181},
      publisher = {Springer-Verlag},
      year = {1982},
      mrclass = {14B25 (14J10 16A18)},
      mrnumber = {0657428},
      mrreviewer = {A. Verschoren},
      address = {New York},
      zblnumber = {0508.14027},
      }
  • [artin-dejong] M. Artin and A. J. de Jong, Stable orders over surfaces, 2003.
    @misc{artin-dejong,
      author = {Artin, M. and de Jong, A. J.},
      title = {Stable orders over surfaces},
      year = {2003},
      note = {preprint},
      }
  • [MR0321934] Go to document M. Artin and D. Mumford, "Some elementary examples of unirational varieties which are not rational," Proc. London Math. Soc., vol. 25, pp. 75-95, 1972.
    @article{MR0321934, mrkey = {0321934},
      author = {Artin, M. and Mumford, D.},
      title = {Some elementary examples of unirational varieties which are not rational},
      journal = {Proc. London Math. Soc.},
      fjournal = {Proceedings of the London Mathematical Society. Third Series},
      volume = {25},
      year = {1972},
      pages = {75--95},
      issn = {0024-6115},
      mrclass = {14J10 (14F25)},
      mrnumber = {0321934},
      mrreviewer = {M. Nagata},
      doi = {10.1112/plms/s3-25.1.75},
      zblnumber = {0244.14017},
      }
  • [ax] Go to document J. Ax, "The elementary theory of finite fields," Ann. of Math., vol. 88, pp. 239-271, 1968.
    @article{ax, mrkey = {0229613},
      author = {Ax, James},
      title = {The elementary theory of finite fields},
      journal = {Ann. of Math.},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {88},
      year = {1968},
      pages = {239--271},
      issn = {0003-486X},
      mrclass = {10.80},
      mrnumber = {0229613},
      mrreviewer = {D. J. Lewis},
      doi = {10.2307/1970573},
      zblnumber = {0195.05701},
      }
  • [cadman_using_2007] Go to document C. Cadman, "Using stacks to impose tangency conditions on curves," Amer. J. Math., vol. 129, iss. 2, pp. 405-427, 2007.
    @article{cadman_using_2007, mrkey = {2306040},
      author = {Cadman, Charles},
      title = {Using stacks to impose tangency conditions on curves},
      journal = {Amer. J. Math.},
      fjournal = {American Journal of Mathematics},
      volume = {129},
      year = {2007},
      number = {2},
      pages = {405--427},
      issn = {0002-9327},
      coden = {AJMAAN},
      mrclass = {14D20 (14A20 14N35)},
      mrnumber = {2306040},
      mrreviewer = {Michael A. Rose},
      doi = {10.1353/ajm.2007.0007},
      zblnumber = {1127.14002},
      }
  • [colliot] J. Colliot-Thélène, Die Brauersche Gruppe; ihre Verallgemeinerungen und Anwendungen in der Arithmetischen Geometrie (Vortragsnotizen, Brauer Tagung, Stuttgart, 22.-24. März 2001), 2001.
    @misc{colliot,
      author = {Colliot-Thélène, Jean-Louis},
      title = {Die {B}rauersche {G}ruppe; ihre {V}erallgemeinerungen und {A}nwendungen in der {A}rithmetischen {G}eometrie ({V}ortragsnotizen, {B}rauer {T}agung, {S}tuttgart, 22.-24. {M}{ä}rz 2001)},
      year = {2001},
      note = {unpublished notes},
      }
  • [MR1923420] Go to document J. Colliot-Thélène, "Exposant et indice d’algèbres simples centrales non ramifiées," Enseign. Math., vol. 48, iss. 1-2, pp. 127-146, 2002.
    @article{MR1923420, mrkey = {1923420},
      author = {Colliot-Th{é}l{è}ne, Jean-Louis},
      title = {Exposant et indice d'algèbres simples centrales non ramifiées},
      note = {with an appendix by Ofer Gabber},
      journal = {Enseign. Math.},
      fjournal = {L'Enseignement Mathématique. Revue Internationale. IIe Série},
      volume = {48},
      year = {2002},
      number = {1-2},
      pages = {127--146},
      issn = {0013-8584},
      coden = {ENMAAR},
      mrclass = {16K20 (14F22)},
      mrnumber = {1923420},
      mrreviewer = {Anne Qu{é}guiner-Mathieu},
      zblnumber = {1047.16007},
      URL = {http://www.unige.ch/math/EnsMath/EM_en/},
     }
  • [dejong-gabber] A. J. de Jong, A result of Gabber, 2003.
    @misc{dejong-gabber,
      author = {de Jong, A. J.},
      title = {A result of {G}abber},
      year = {2003},
      note = {preprint},
      }
  • [MR2060023] Go to document A. J. de Jong, "The period-index problem for the Brauer group of an algebraic surface," Duke Math. J., vol. 123, iss. 1, pp. 71-94, 2004.
    @article{MR2060023, mrkey = {2060023},
      author = {de Jong, A. J.},
      title = {The period-index problem for the {B}rauer group of an algebraic surface},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {123},
      year = {2004},
      number = {1},
      pages = {71--94},
      issn = {0012-7094},
      coden = {DUMJAO},
      mrclass = {14F22},
      mrnumber = {2060023},
      mrreviewer = {Stefan Schr{ö}er},
      doi = {10.1215/S0012-7094-04-12313-9},
      zblnumber = {1060.14025},
      }
  • [djs] Go to document J. Starr and J. de Jong, "Almost proper GIT-stacks and discriminant avoidance," Doc. Math., vol. 15, pp. 957-972, 2010.
    @article{djs, mrkey = {2745688},
      author = {Starr, Jason and de Jong, Johan},
      title = {Almost proper {GIT}-stacks and discriminant avoidance},
      journal = {Doc. Math.},
      fjournal = {Documenta Mathematica},
      volume = {15},
      year = {2010},
      pages = {957--972},
      issn = {1431-0635},
      mrclass = {14J10 (14L15)},
      mrnumber = {2745688},
      zblnumber = {1210.14013},
      url = {http://www.math.uiuc.edu/documenta/vol-15/29.html},
      }
  • [kresch-cycle] Go to document A. Kresch, "Cycle groups for Artin stacks," Invent. Math., vol. 138, iss. 3, pp. 495-536, 1999.
    @article{kresch-cycle, mrkey = {1719823},
      author = {Kresch, Andrew},
      title = {Cycle groups for {A}rtin stacks},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {138},
      year = {1999},
      number = {3},
      pages = {495--536},
      issn = {0020-9910},
      coden = {INVMBH},
      mrclass = {14A20 (14C17)},
      mrnumber = {1719823},
      mrreviewer = {Burt Totaro},
      doi = {10.1007/s002220050351},
      zblnumber = {0938.14003},
      }
  • [MR2026412] Go to document A. Kresch and A. Vistoli, "On coverings of Deligne—Mumford stacks and surjectivity of the Brauer map," Bull. London Math. Soc., vol. 36, iss. 2, pp. 188-192, 2004.
    @article{MR2026412, mrkey = {2026412},
      author = {Kresch, Andrew and Vistoli, Angelo},
      title = {On coverings of {D}eligne—{M}umford stacks and surjectivity of the {B}rauer map},
      journal = {Bull. London Math. Soc.},
      fjournal = {The Bulletin of the London Mathematical Society},
      volume = {36},
      year = {2004},
      number = {2},
      pages = {188--192},
      issn = {0024-6093},
      coden = {LMSBBT},
      mrclass = {14A20 (14D20 14F22)},
      mrnumber = {2026412},
      mrreviewer = {Martin M{ö}ller},
      doi = {10.1112/S0024609303002728},
      zblnumber = {1062.14004},
      }
  • [orbi] Go to document M. Lieblich, "Moduli of twisted orbifold sheaves," Adv. Math., vol. 226, iss. 5, pp. 4145-4182, 2011.
    @article{orbi, mrkey = {2770444},
      author = {Lieblich, Max},
      title = {Moduli of twisted orbifold sheaves},
      journal = {Adv. Math.},
      fjournal = {Advances in Mathematics},
      volume = {226},
      year = {2011},
      number = {5},
      pages = {4145--4182},
      issn = {0001-8708},
      coden = {ADMTA4},
      mrclass = {14D23},
      mrnumber = {2770444},
      mrreviewer = {Hsian-Hua Tseng},
      doi = {10.1016/j.aim.2010.11.011},
      zblnumber = {1211.14050},
      }
  • [paiitbgoaas] Go to document M. Lieblich, "Period and index in the Brauer group of an arithmetic surface," J. Reine Angew. Math., vol. 659, pp. 1-41, 2011.
    @article{paiitbgoaas, mrkey = {2837009},
      author = {Lieblich, Max},
      title = {Period and index in the {B}rauer group of an arithmetic surface},
      note = {with an appendix by Daniel Krashen},
      journal = {J. Reine Angew. Math.},
      fjournal = {Journal für die Reine und Angewandte Mathematik. [Crelle's Journal]},
      volume = {659},
      year = {2011},
      pages = {1--41},
      issn = {0075-4102},
      coden = {JRMAA8},
      mrclass = {14F22 (14G40)},
      mrnumber = {2837009},
      mrreviewer = {Stefan Schr{ö}er},
      doi = {10.1515/CRELLE.2011.059},
      zblnumber = {1230.14021},
      }
  • [twisted-moduli] Go to document M. Lieblich, "Moduli of twisted sheaves," Duke Math. J., vol. 138, iss. 1, pp. 23-118, 2007.
    @article{twisted-moduli, mrkey = {2309155},
      author = {Lieblich, Max},
      title = {Moduli of twisted sheaves},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {138},
      year = {2007},
      number = {1},
      pages = {23--118},
      issn = {0012-7094},
      coden = {DUMJAO},
      mrclass = {14D20 (14A20 14F05)},
      mrnumber = {2309155},
      mrreviewer = {Nicolas Perrin},
      doi = {10.1215/S0012-7094-07-13812-2},
      zblnumber = {1122.14012},
      }
  • [period-index-paper] Go to document M. Lieblich, "Twisted sheaves and the period-index problem," Compos. Math., vol. 144, iss. 1, pp. 1-31, 2008.
    @article{period-index-paper, mrkey = {2388554},
      author = {Lieblich, Max},
      title = {Twisted sheaves and the period-index problem},
      journal = {Compos. Math.},
      fjournal = {Compositio Mathematica},
      volume = {144},
      year = {2008},
      number = {1},
      pages = {1--31},
      issn = {0010-437X},
      mrclass = {14F22 (14A20 14D20 16K50)},
      mrnumber = {2388554},
      mrreviewer = {Stefan Schr{ö}er},
      doi = {10.1112/S0010437X07003144},
      zblnumber = {1133.14018},
      }
  • [matzri] Go to document E. Matzri, "Symbol length in the Brauer group of a field," Trans. Amer. Math. Soc..
    @article{matzri,
      author = {Matzri, Eli},
      title = {Symbol length in the {B}rauer group of a field},
      journal = {Trans. Amer. Math. Soc.},
      note={published electronically: April 15, 2015},
      doi = {10.1090/tran/6326},
      }
  • [MR1432058] Go to document L. Moret-Bailly, "Un problème de descente," Bull. Soc. Math. France, vol. 124, iss. 4, pp. 559-585, 1996.
    @article{MR1432058, mrkey = {1432058},
      author = {Moret-Bailly, Laurent},
      title = {Un problème de descente},
      journal = {Bull. Soc. Math. France},
      fjournal = {Bulletin de la Société Mathématique de France},
      volume = {124},
      year = {1996},
      number = {4},
      pages = {559--585},
      issn = {0037-9484},
      coden = {BSMFAA},
      mrclass = {14A20 (13F40 13J10)},
      mrnumber = {1432058},
      mrreviewer = {D.-M. Popescu},
      zblnumber = {0914.13002},
      url = {http://smf4.emath.fr/Publications/Bulletin/124/html/smf_bull_124_559-585.html},
      }
  • [MR1376250] Go to document K. G. O’Grady, "Moduli of vector bundles on projective surfaces: some basic results," Invent. Math., vol. 123, iss. 1, pp. 141-207, 1996.
    @article{MR1376250, mrkey = {1376250},
      author = {O'Grady, Kieran G.},
      title = {Moduli of vector bundles on projective surfaces: some basic results},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {123},
      year = {1996},
      number = {1},
      pages = {141--207},
      issn = {0020-9910},
      coden = {INVMBH},
      mrclass = {14D20},
      mrnumber = {1376250},
      mrreviewer = {Jean-Marc Drezet},
      doi = {10.1007/BF01232371},
      zblnumber = {0869.14005},
      }
  • [MR2183251] Go to document M. C. Olsson, "On proper coverings of Artin stacks," Adv. Math., vol. 198, iss. 1, pp. 93-106, 2005.
    @article{MR2183251, mrkey = {2183251},
      author = {Olsson, Martin C.},
      title = {On proper coverings of {A}rtin stacks},
      journal = {Adv. Math.},
      fjournal = {Advances in Mathematics},
      volume = {198},
      year = {2005},
      number = {1},
      pages = {93--106},
      issn = {0001-8708},
      coden = {ADMTA4},
      mrclass = {14A20},
      mrnumber = {2183251},
      mrreviewer = {Charles D. Cadman},
      doi = {10.1016/j.aim.2004.08.017},
      zblnumber = {1084.14004},
      }
  • [MR1462850] Go to document D. J. Saltman, "Division algebras over $p$-adic curves," J. Ramanujan Math. Soc., vol. 12, iss. 1, pp. 25-47, 1997.
    @article{MR1462850, mrkey = {1462850},
      author = {Saltman, David J.},
      title = {Division algebras over {$p$}-adic curves},
      journal = {J. Ramanujan Math. Soc.},
      fjournal = {Journal of the Ramanujan Mathematical Society},
      volume = {12},
      year = {1997},
      number = {1},
      pages = {25--47},
      issn = {0970-1249},
      mrclass = {16H05 (12E15 13A20 16K20)},
      mrnumber = {1462850},
      mrreviewer = {Timothy J. Ford},
      zblnumber = {0902.16021},
      url = {https://drive.google.com/file/d/0B7eMKhVCM8ijRzJTWHRUMGdrXzA/edit},
      }
  • [saltman_cyclic_2007] Go to document D. J. Saltman, "Cyclic algebras over $p$-adic curves," J. Algebra, vol. 314, iss. 2, pp. 817-843, 2007.
    @article{saltman_cyclic_2007, mrkey = {2344586},
      author = {Saltman, David J.},
      title = {Cyclic algebras over {$p$}-adic curves},
      journal = {J. Algebra},
      fjournal = {Journal of Algebra},
      volume = {314},
      year = {2007},
      number = {2},
      pages = {817--843},
      issn = {0021-8693},
      coden = {JALGA4},
      mrclass = {16K20 (11S15 16K50)},
      mrnumber = {2344586},
      mrreviewer = {Jan van Geel},
      doi = {10.1016/j.jalgebra.2007.03.003},
      zblnumber = {1129.16014},
      }
  • [MR554237] . J-P. Serre, Local Fields, New York: Springer-Verlag, 1979, vol. 67.
    @book{MR554237, mrkey = {0554237},
      author = {Serre, {\relax J-P}},
      title = {Local {F}ields},
      series = {Grad. Texts in Math.},
      volume = {67},
      note = {translated from the French by Marvin Jay Greenberg},
      publisher = {Springer-Verlag},
      year = {1979},
      pages = {viii+241},
      isbn = {0-387-90424-7},
      mrclass = {12Bxx},
      mrnumber = {0554237},
      address = {New York},
      zblnumber = {0423.12016},
      }

Authors

Max Lieblich

University of Washington, Seattle, WA