Algebras and Representation Theory

, Volume 17, Issue 6, pp 1635–1655 | Cite as

An Isomorphism Problem for Azumaya Algebras with Involution over Semilocal Bézout Domains

Article
  • 105 Downloads

Abstract

Let R be a semilocal Bézout domain with fraction field F. Assume that 2 is invertible in R. The main result of this article states that R−algebras with involution that become isomorphic over F are already isomorphic over R. We also show that this implies that hermitian or skew-hermitian spaces over an R−algebra with involution without zero divisors that become similar over F, are already similar over R.

Keywords

Azumaya algebras with involution Central simple algebras with involution Algebraic groups Valuation rings Semilocal Bézout domains (skew-)hermitian spaces Bilinear spaces Multipliers 

Mathematics Subject Classifications (2010)

16H05 16K20 12J20 20G35 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Auslander, M., Goldman, O.: The Brauer group of a commutative ring. Trans. Amer. Math. Soc. 97, 367–409 (1960)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Beauregard, R.A.: Overrings of Bézout domains. Canad. Math. Bull. 16, 475–477 (1973)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Beke, S.: Specialisation and good reduction for algebras with involution. preprint, http://www.math.uni-bielefeld.de/lag/man/488.pdf (2013)
  4. 4.
    Chase, S., Harrison, D., Rosenberg, A.: Galois theory and galois cohomology of commutative rings. Mem. Amer. Math. Soc. 52, 15–33 (1965)MATHMathSciNetGoogle Scholar
  5. 5.
    Elman, R., Karpenko, N.A., Merkurjev, A.S.: The algebraic and geometric theory of quadratic forms, Colloq. Publ., Am. Math. Soc., vol. 56, Am. Math. Soc. (2008)Google Scholar
  6. 6.
    Engler, A., Prestel, A.: Valued fields. Springer Monographs in Mathematics. Springer, Berlin Heidelberg New York (2005)Google Scholar
  7. 7.
    Fuchs, L., Salce, L.: Modules over non-Noetherian domains. Mathematical Surveys and Monographs, vol. 84. American Mathematical Society, Providence (2001)Google Scholar
  8. 8.
    Grothendieck, A.: La torsion homologique et les sections rationnelles. In: Anneaux de Chow et applications, Séminaire C. Chevalley, 2e année, Secrétariat mathématique (1958)Google Scholar
  9. 9.
    Grothendieck, A.: Le groupe de Brauer II. In: Dix exposés sur la cohomologie des schémas, Advanced Studies in Pure Mathematics (1968)Google Scholar
  10. 10.
    Harder, G.: Eine Bemerkung zum schwachen Approximationssatz. Arch. Math. (Basel) 19, 465–471 (1968)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Hinohara, Y.: Projective modules over semilocal rings. Tôhoku Math. J. 14(2), 205–211 (1962)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Knus, M.-A.: Quadratic and hermitian forms over rings. Grundlehren der mathematischen Wissenschaften, vol. 294. Springer, Berlin Heidelberg New York (1991)CrossRefGoogle Scholar
  13. 13.
    Knus, M.-A., Merkurjev, A.S., Rost, M., Tignol, J.-P.: The book of involutions, Colloq. Publ., Am. Math. Soc., vol. 44, Am. Math. Soc. (1998)Google Scholar
  14. 14.
    Lang, S.: Algebra. Revised Third edition. Graduate Texts in Mathematics, Springer, New York, Inc. (2002)Google Scholar
  15. 15.
    Nisnevich, Y.: Rationally trivial principal homogeneous spaces, purity and arithmetic of reductive group schemes over extensions of two-dimensional regular local rings. C. R. Acad. Sci. Paris Sér. I Math. 309(10), 651–655 (1989)MATHMathSciNetGoogle Scholar
  16. 16.
    Panin, I.A.: Purity for multipliers. Algebra and number theory, pp. 66–89. Hindustan Book Agency, Delhi (2005)Google Scholar
  17. 17.
    Ribenboim, P.: Le théorème d’approximation pour les valuations de Krull. Math. Z. 68, 1–18 (1957)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Saltman, D.J.: Azumaya algebras with involution. J. Algebra 52(2), 526–539 (1978)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Serre, J.-P.: Les espaces fibrés algébriques. In: Anneaux de Chow et applications, Séminaire C. Chevalley; 2e année, Secrétariat mathématique (1958)Google Scholar
  20. 20.
    Tignol, J.-P.: A Cassels-Pfister theorem for involutions on central simple algebras. J. Algebra 181(3), 857–875 (1996)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of MathematicsGhent UniversityGhentBelgium

Personalised recommendations