Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Formal constructions in the Brauer group of the function field of a $ p$-adic curve


Authors: Eric Brussel and Eduardo Tengan
Journal: Trans. Amer. Math. Soc. 367 (2015), 3299-3321
MSC (2010): Primary 11G20, 11R58, 14E22, 16K50
DOI: https://doi.org/10.1090/S0002-9947-2014-06154-0
Published electronically: December 19, 2014
MathSciNet review: 3314809
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study the relationship between the cohomology of the function field of a curve over a complete discretely valued field and that of the function ring of curves resulting over its residue field. The results are applied to prove the existence of noncrossed product division algebras and indecomposable division algebras of unequal period and index over the function field of any $ p$-adic curve, generalizing the results and methods of a previous work of the authors and McKinnie.


References [Enhancements On Off] (What's this?)

  • [1] S. A. Amitsur, On central division algebras, Israel J. Math. 12 (1972), 408-420. MR 0318216 (47 #6763)
  • [2] S. A. Amitsur, Division algebras. A survey, Algebraists' homage: papers in ring theory and related topics (New Haven, Conn., 1981), Contemp. Math., vol. 13, Amer. Math. Soc., Providence, R.I., 1982, pp. 3-26. MR 685935 (84b:16021)
  • [3] S. A. Amitsur, L. H. Rowen, and J.-P. Tignol, Division algebras of degree $ 4$ and $ 8$ with involution, Israel J. Math. 33 (1979), no. 2, 133-148. MR 571249 (81h:16029), https://doi.org/10.1007/BF02760554
  • [4] M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0242802 (39 #4129)
  • [5] Asher Auel, Eric Brussel, Skip Garibaldi, and Uzi Vishne, Open problems on central simple algebras, Transform. Groups 16 (2011), no. 1, 219-264. MR 2785502 (2012e:16050), https://doi.org/10.1007/s00031-011-9119-8
  • [6] Eric Brussel, Noncrossed products and nonabelian crossed products over $ \mathbf {Q}(t)$ and $ \mathbf {Q}((t))$, Amer. J. Math. 117 (1995), no. 2, 377-393. MR 1323680 (96a:16014), https://doi.org/10.2307/2374919
  • [7] Eric S. Brussel, Decomposability and embeddability of discretely Henselian division algebras. part A, Israel J. Math. 96 (1996), no. part A, 141-183. MR 1432730 (97m:16030), https://doi.org/10.1007/BF02785537
  • [8] Eric S. Brussel, Noncrossed products over $ k_{\mathfrak{p}}(t)$, Trans. Amer. Math. Soc. 353 (2001), no. 5, 2115-2129 (electronic). MR 1813610 (2001j:16022), https://doi.org/10.1090/S0002-9947-00-02626-X
  • [9] Eric Brussel, On Saltman's $ p$-adic curves papers, Quadratic forms, linear algebraic groups, and cohomology, Dev. Math., vol. 18, Springer, New York, 2010, pp. 13-39. MR 2648718 (2011k:16041), https://doi.org/10.1007/978-1-4419-6211-9_2
  • [10] E. Brussel, K. McKinnie, and E. Tengan, Indecomposable and noncrossed product division algebras over function fields of smooth $ p$-adic curves, Adv. Math. 226 (2011), no. 5, 4316-4337. MR 2770451 (2012g:14033), https://doi.org/10.1016/j.aim.2010.12.005
  • [11] E. Brussel and E. Tengan,
    Division algebras of prime period $ \ell \neq p$ over function fields of $ p$-adic curves,
    Israel J. Math. (to appear).
  • [12] J.-L. Colliot-Thélène, Birational invariants, purity and the Gersten conjecture, division algebras (Santa Barbara, CA, 1992) Proc. Sympos. Pure Math., vol. 58, Amer. Math. Soc., Providence, RI, 1995, pp. 1-64. MR 1327280 (96c:14016)
  • [13] Eberhard Freitag and Reinhardt Kiehl, Étale cohomology and the Weil conjecture, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 13, Springer-Verlag, Berlin, 1988. Translated from the German by Betty S. Waterhouse and William C. Waterhouse; With an historical introduction by J. A. Dieudonné. MR 926276 (89f:14017)
  • [14] László Fuchs, Infinite abelian groups. Vol. I, Pure and Applied Mathematics, Vol. 36, Academic Press, New York, 1970. MR 0255673 (41 #333)
  • [15] Kazuhiro Fujiwara, A proof of the absolute purity conjecture (after Gabber), Algebraic geometry 2000, Azumino (Hotaka), Adv. Stud. Pure Math., vol. 36, Math. Soc. Japan, Tokyo, 2002, pp. 153-183. MR 1971516 (2004d:14015)
  • [16] Skip Garibaldi, Alexander Merkurjev, and Jean-Pierre Serre, Cohomological invariants in Galois cohomology, University Lecture Series, vol. 28, American Mathematical Society, Providence, RI, 2003. MR 1999383 (2004f:11034)
  • [17] A. Grothendieck, Éléments de géométrie algébrique. I. Le langage des schémas, Inst. Hautes Études Sci. Publ. Math. 4 (1960), 228. MR 0217083 (36 #177a)
  • [18] A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II, Inst. Hautes Études Sci. Publ. Math. 24 (1965), 231 (French). MR 0199181 (33 #7330)
  • [19] A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III, Inst. Hautes Études Sci. Publ. Math. 28 (1966), 255. MR 0217086 (36 #178)
  • [20] A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV, Inst. Hautes Études Sci. Publ. Math. 32 (1967), 361 (French). MR 0238860 (39 #220)
  • [21] A. Grothendieck, Revêtements étales et groupe fondamental (SGA 1), Documents Mathématiques (Paris) [Mathematical Documents (Paris)], 3, Société Mathématique de France, Paris, 2003 (French). Séminaire de géométrie algébrique du Bois Marie 1960-61. [Algebraic Geometry Seminar of Bois Marie 1960-61]; Directed by A. Grothendieck; With two papers by M. Raynaud; Updated and annotated reprint of the 1971 original [Lecture Notes in Math., 224, Springer, Berlin; MR0354651 (50 #7129)]. MR 2017446 (2004g:14017)
  • [22] Alexander Grothendieck and Jacob P. Murre, The tame fundamental group of a formal neighbourhood of a divisor with normal crossings on a scheme, Lecture Notes in Mathematics, Vol. 208, Springer-Verlag, Berlin, 1971. MR 0316453 (47 #5000)
  • [23] David Harbater, Julia Hartmann, and Daniel Krashen, Applications of patching to quadratic forms and central simple algebras, Invent. Math. 178 (2009), no. 2, 231-263. MR 2545681 (2010j:11058), https://doi.org/10.1007/s00222-009-0195-5
  • [24] David Harbater, Julia Hartmann, and Daniel Krashen, Patching subfields of division algebras, Trans. Amer. Math. Soc. 363 (2011), no. 6, 3335-3349. MR 2775810 (2012c:12008), https://doi.org/10.1090/S0002-9947-2010-05229-8
  • [25] Bill Jacob, Indecomposable division algebras of prime exponent, J. Reine Angew. Math. 413 (1991), 181-197. MR 1089801 (91m:12002), https://doi.org/10.1515/crll.1991.413.181
  • [26] N. Karpenko, Chow groups of quadrics and index reduction formula, Nova J. Algebra Geom. 3 (1995), no. 4, 357-379. MR 1341100 (96e:14003)
  • [27] Nikita A. Karpenko, Codimension $ 2$ cycles on Severi-Brauer varieties, $ K$-Theory 13 (1998), no. 4, 305-330. MR 1615533 (99b:16030), https://doi.org/10.1023/A:1007705720373
  • [28] Kazuya Kato, A Hasse principle for two-dimensional global fields, J. Reine Angew. Math. 366 (1986), 142-183. With an appendix by Jean-Louis Colliot-Thélène. MR 833016 (88b:11036), https://doi.org/10.1515/crll.1986.366.142
  • [29] Serge Lang, Algebra, 3rd ed., Graduate Texts in Mathematics, vol. 211, Springer-Verlag, New York, 2002. MR 1878556 (2003e:00003)
  • [30] Max Lieblich, Period and index in the Brauer group of an arithmetic surface, J. Reine Angew. Math. 659 (2011), 1-41. With an appendix by Daniel Krashen. MR 2837009, https://doi.org/10.1515/CRELLE.2011.059
  • [31] 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 (2003g:14001)
  • [32] Hideyuki Matsumura, Commutative ring theory, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1989. Translated from the Japanese by M. Reid. MR 1011461 (90i:13001)
  • [33] James S. Milne, Étale cohomology, Princeton Mathematical Series, vol. 33, Princeton University Press, Princeton, N.J., 1980. MR 559531 (81j:14002)
  • [34] Raman Parimala and V. Suresh, The $ u$-invariant of the function fields of $ p$-adic curves, Ann. of Math. (2) 172 (2010), no. 2, 1391-1405. MR 2680494 (2011g:11074), https://doi.org/10.4007/annals.2010.172.1397
  • [35] Wayne Raskind, Abelian class field theory of arithmetic schemes, division algebras (Santa Barbara, CA, 1992) Proc. Sympos. Pure Math., vol. 58, Amer. Math. Soc., Providence, RI, 1995, pp. 85-187. MR 1327282 (96b:11089)
  • [36] Shuji Saito, Class field theory for curves over local fields, J. Number Theory 21 (1985), no. 1, 44-80. MR 804915 (87g:11075), https://doi.org/10.1016/0022-314X(85)90011-3
  • [37] David J. Saltman, Indecomposable division algebras, Comm. Algebra 7 (1979), no. 8, 791-817. MR 529494 (80g:16021), https://doi.org/10.1080/00927877908822376
  • [38] David J. Saltman, Finite-dimensional division algebras, Azumaya algebras, actions, and modules (Bloomington, IN, 1990) Contemp. Math., vol. 124, Amer. Math. Soc., Providence, RI, 1992, pp. 203-214. MR 1144037 (93a:16014), https://doi.org/10.1090/conm/124/1144037
  • [39] David J. Saltman, Division algebras over $ p$-adic curves, J. Ramanujan Math. Soc. 12 (1997), no. 1, 25-47. MR 1462850 (98d:16032)
  • [40] David J. Saltman, Correction to: ``Division algebras over $ p$-adic curves'' [J. Ramanujan Math. Soc. 12 (1997), no. 1, 25-47; MR1462850 (98d:16032)], J. Ramanujan Math. Soc. 13 (1998), no. 2, 125-129. MR 1666362 (99k:16036)
  • [41] David J. Saltman, Cyclic algebras over $ p$-adic curves, J. Algebra 314 (2007), no. 2, 817-843. MR 2344586 (2008i:16018), https://doi.org/10.1016/j.jalgebra.2007.03.003
  • [42] Venapally Suresh, Bounding the symbol length in the Galois cohomology of function fields of $ p$-adic curves, Comment. Math. Helv. 85 (2010), no. 2, 337-346. MR 2595182 (2011b:12010), https://doi.org/10.4171/CMH/198

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 11G20, 11R58, 14E22, 16K50

Retrieve articles in all journals with MSC (2010): 11G20, 11R58, 14E22, 16K50


Additional Information

Eric Brussel
Affiliation: Department of Mathematics, California Polytechnic State University, San Luis Obispo, California 93407
Email: ebrussel@calpoly.edu

Eduardo Tengan
Affiliation: Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, São Carlos, São Paulo, Brazil
Email: etengan@icmc.usp.br

DOI: https://doi.org/10.1090/S0002-9947-2014-06154-0
Received by editor(s): March 21, 2012
Received by editor(s) in revised form: April 11, 2013
Published electronically: December 19, 2014
Additional Notes: The second author was supported by CNPq grant 303817/2011-9.
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society