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SK1 of Azumaya algebras over Hensel Pairs

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Abstract

Let A be an Azumaya algebra of constant rank n 2 over a Hensel pair (R, I) where R is a semilocal ring with n invertible in R. Then the reduced Whitehead group SK1(A) coincides with its reduction SK1(A/I A). This generalizes a result of Hazrat (J Algebra 305:687–703, 2006) to non-local Henselian rings.

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References

  1. Bourbaki N.: Bourbaki, Commutative Algebra, Chap. 1–7. Springer, New York (1989)

    Google Scholar 

  2. Goldman O.: Goldman, determinants in projective modules. Nagoya Math. J. 3, 7–11 (1966)

    Google Scholar 

  3. Greco S.: Algebras over nonlocal Hensel rings. J. Algebra 8, 45–59 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  4. Greco S.: Algebras over nonlocal Hensel rings II. J. Algebra 13, 48–56 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  5. Grothendieck A.: Le groupe de Brauer. III: Dix exposés la cohomologie des schémas. North Holland, Amsterdam (1968)

    Google Scholar 

  6. Hazrat R.: Reduced K-theory of Azumaya algebras. J. Algebra 305, 687–703 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  7. Hazrat R.: On the first Galois cohomology group of the algebraic group SL1(D). Comm. Algebra 36, 381–387 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  8. Jacob B., Wadsworth A.: Division algebras over Henselian fields. J. Algebra 128, 126–179 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  9. Lam T.Y.: A First Course in Noncommutative Rings. Springer, New York (1991)

    MATH  Google Scholar 

  10. Knus M.-A.: Quadratic and Hermitian Forms Over Rings. Springer, Berlin (1991)

    MATH  Google Scholar 

  11. Platonov VP: The Tannaka–Artin problem and reduced K-theory. Math. USSR Izv. 10, 211–243 (1976)

    Article  Google Scholar 

  12. Raynaud, M.: Anneaux locaux Hensèliens. LNM, vol. 169. Springer, Heidelberg (1070)

  13. Rosenberg, J.: Algebraic K-theory and its Applications, GTM, vol. 147. Springer, Heidelberg (1994)

  14. Saltman, D.: Lectures on division algebras. RC Series in Mathematics, vol. 94. AMS, Providence (1999)

  15. Strano R.: Principal homogenous spaces over Hensel rings. Proc. Am. Math. Soc. 87(2), 208–212 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  16. Suslin, A.: SK1 of division algebras and Galois cohomology. Adv. Soviet Math., vol. 4, pp. 75–99. AMS, Providence (1991)

  17. Vaserstein L.: On the Whitehead determinant for semilocal rings. J. Algebra 283, 690–699 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  18. Wadsworth, A.: Valuation theory on finite dimensional division algebras. Fields Inst. Commun., vol. 32, pp. 385–449. American Mathematical Society, Providence (2002)

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  1. Department of Pure Mathematics, Queen’s University, Belfast, BT7 1NN, UK

    Roozbeh Hazrat

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  1. Roozbeh Hazrat
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Correspondence to Roozbeh Hazrat.

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Hazrat, R. SK1 of Azumaya algebras over Hensel Pairs. Math. Z. 264, 295–299 (2010). https://doi.org/10.1007/s00209-008-0464-9

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  • DOI: https://doi.org/10.1007/s00209-008-0464-9

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