manuscripta mathematica

, Volume 130, Issue 3, pp 387–409 | Cite as

Singularities of generic characteristic polynomials and smooth finite splittings of Azumaya algebras over surfaces

Open Access


Let k be an algebraically closed field. Let P(X 11, . . . , X nn , T) be the characteristic polynomial of the generic matrix (X ij ) over k. We determine its singular locus as well as the singular locus of its Galois splitting. If X is a smooth quasi-projective surface over k and A an Azumaya algebra on X of degree n, using a method suggested by M. Artin, we construct finite smooth splittings for A of degree n over X whose Galois closures are smooth.

Mathematics Subject Classification (2000)

Primary 16H05 Secondary 14F22 



The authors acknowledge support from NSF DMS-0653382 and from Max-Planck Institut für Mathematik, Bonn. We thank Jean-Louis Colliot-Thélène, Aise Johan de Jong, and David Saltman for several discussions.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.


  1. 1.
    Bourbaki N.: Algèbre, Chapitre IV, Polynômes et fractions rationnelles. Masson, Paris (1981)Google Scholar
  2. 2.
    Colliot-Thélène J.-L.: Algèbres simples centrales sur les corps de fonctions de deux variables (d’après A. J. de Jong), Séminaire Bourbaki, juin 2005, Exp. No. 949. Astérisque 307, 379–413 (2006)Google Scholar
  3. 3.
    Ekedahl T., Laksov D.: Splitting algebras, symmetric functions and Galois theory. J. Algebra Appl. 4, 59–75 (2004)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Fried M.D., Jarden M.: Field Arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer, New York (2005)Google Scholar
  5. 5.
    Grothendieck A.: Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : IV. Étude locale des schémas et des morphismes de schémas, Troisième partie. Publications Mathématiques de l’IHÉS 28, 5–255 (1966)Google Scholar
  6. 6.
    Hartshorne R.: Algebraic Geometry, Graduate Texts in Mathematics, vol. 52. Springer-Verlag, New York (1977)Google Scholar
  7. 7.
    Jantzen, J.C.: Nilpotent orbits in representation theory. In: Ørsted, B., Anker, J.-Ph. (eds.) Lie Theory: Lie Algebras and Representations, pp. 1–211. Birkhäuser, Boston (2004)Google Scholar
  8. 8.
    Van den Bergh, M.: Notes on de Jong’s period=index theorem for central simple algebras over fields of transcendence degree two. Preprint (2007).

Copyright information

© The Author(s) 2009

Authors and Affiliations

  1. 1.IGATEPFLLausanneSwitzerland
  2. 2.Department of Mathematics and Computer ScienceEmory UniversityAtlantaUSA

Personalised recommendations