Formal constructions in the Brauer group of the function field of a $p$-adic curve
HTML articles powered by AMS MathViewer
- by Eric Brussel and Eduardo Tengan PDF
- Trans. Amer. Math. Soc. 367 (2015), 3299-3321 Request permission
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
- S. A. Amitsur, On central division algebras, Israel J. Math. 12 (1972), 408–420. MR 318216, DOI 10.1007/BF02764632
- 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
- 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, DOI 10.1007/BF02760554
- M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0242802
- 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, DOI 10.1007/s00031-011-9119-8
- 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, DOI 10.2307/2374919
- 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, DOI 10.1007/BF02785537
- Eric S. Brussel, Noncrossed products over $k_{\mathfrak {p}}(t)$, Trans. Amer. Math. Soc. 353 (2001), no. 5, 2115–2129. MR 1813610, DOI 10.1090/S0002-9947-00-02626-X
- 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, DOI 10.1007/978-1-4419-6211-9_{2}
- 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, DOI 10.1016/j.aim.2010.12.005
- 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).
- J.-L. Colliot-Thélène, Birational invariants, purity and the Gersten conjecture, $K$-theory and algebraic geometry: connections with quadratic forms and division algebras (Santa Barbara, CA, 1992) Proc. Sympos. Pure Math., vol. 58, Amer. Math. Soc., Providence, RI, 1995, pp. 1–64. MR 1327280
- 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, DOI 10.1007/978-3-662-02541-3
- László Fuchs, Infinite abelian groups. Vol. I, Pure and Applied Mathematics, Vol. 36, Academic Press, New York-London, 1970. MR 0255673
- 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, DOI 10.2969/aspm/03610153
- 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, DOI 10.1090/ulect/028
- A. Grothendieck, Éléments de géométrie algébrique. I. Le langage des schémas, Inst. Hautes Études Sci. Publ. Math. 4 (1960), 228 (French). MR 217083
- 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 199181
- 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 217086
- 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 238860
- Revêtements étales et groupe fondamental (SGA 1), Documents Mathématiques (Paris) [Mathematical Documents (Paris)], vol. 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
- 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-New York, 1971. MR 0316453
- 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, DOI 10.1007/s00222-009-0195-5
- David Harbater, Julia Hartmann, and Daniel Krashen, Patching subfields of division algebras, Trans. Amer. Math. Soc. 363 (2011), no. 6, 3335–3349. MR 2775810, DOI 10.1090/S0002-9947-2010-05229-8
- Bill Jacob, Indecomposable division algebras of prime exponent, J. Reine Angew. Math. 413 (1991), 181–197. MR 1089801, DOI 10.1515/crll.1991.413.181
- N. Karpenko, Chow groups of quadrics and index reduction formula, Nova J. Algebra Geom. 3 (1995), no. 4, 357–379. MR 1341100
- Nikita A. Karpenko, Codimension $2$ cycles on Severi-Brauer varieties, $K$-Theory 13 (1998), no. 4, 305–330. MR 1615533, DOI 10.1023/A:1007705720373
- 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, DOI 10.1515/crll.1986.366.142
- Serge Lang, Algebra, 3rd ed., Graduate Texts in Mathematics, vol. 211, Springer-Verlag, New York, 2002. MR 1878556, DOI 10.1007/978-1-4613-0041-0
- 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, DOI 10.1515/CRELLE.2011.059
- 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
- 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
- James S. Milne, Étale cohomology, Princeton Mathematical Series, No. 33, Princeton University Press, Princeton, N.J., 1980. MR 559531
- 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, DOI 10.4007/annals.2010.172.1397
- Wayne Raskind, Abelian class field theory of arithmetic schemes, $K$-theory and algebraic geometry: connections with quadratic forms and division algebras (Santa Barbara, CA, 1992) Proc. Sympos. Pure Math., vol. 58, Amer. Math. Soc., Providence, RI, 1995, pp. 85–187. MR 1327282
- Shuji Saito, Class field theory for curves over local fields, J. Number Theory 21 (1985), no. 1, 44–80. MR 804915, DOI 10.1016/0022-314X(85)90011-3
- David J. Saltman, Indecomposable division algebras, Comm. Algebra 7 (1979), no. 8, 791–817. MR 529494, DOI 10.1080/00927877908822376
- 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, DOI 10.1090/conm/124/1144037
- David J. Saltman, Division algebras over $p$-adic curves, J. Ramanujan Math. Soc. 12 (1997), no. 1, 25–47. MR 1462850
- 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
- David J. Saltman, Cyclic algebras over $p$-adic curves, J. Algebra 314 (2007), no. 2, 817–843. MR 2344586, DOI 10.1016/j.jalgebra.2007.03.003
- 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, DOI 10.4171/CMH/198
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
- 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.
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3314809