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Comparing Invariants of SK1

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Abstract

In this text, we compare an invariant of the reduced Whitehead group SK 1 of a central simple algebra recently introduced by Kahn (2010) to other invariants of SK 1. Doing so, we prove the non-triviality of Kahn’s invariant using the non-triviality of an invariant introduced by Suslin (1991) which is non-trivial for Platonov’s examples of non-trivial SK 1 (Platonov, Math USSR Izv 10(2):211–243, 1976). We also give a formula for the value on the centre of the tensor product of two symbol algebras which generalises a formula of Merkurjev for biquaternion algebras (Merkurjev 1995).

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References

  1. Auslander, M., Goldman, O.: The Brauer group of a commutative ring. Trans. Am. Math. Soc. 97, 367–409 (1960)

    Article  MathSciNet  Google Scholar 

  2. Blanchet, A.: Function fields of generalized Brauer–Severi varieties. Commun. Algebra 19(1), 97–118 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cohen, I.: On the structure and ideal theory of complete local rings. Trans. Am. Math. Soc. 59, 54–106 (1946)

    Article  MATH  Google Scholar 

  4. Draxl, P.: Skew fields. In: London Mathematical Society Lecture Note Series, vol. 81. Cambridge University Press, Cambridge (1983)

    Google Scholar 

  5. Garibaldi, S.: Cohomological invariants: exceptional groups and spin groups. Mem. Am. Math. Soc. 200(937), xii+81 (2009) (With an appendix by Detlev W. Hoffmann)

  6. Garibaldi, S., Merkurjev, A., Serre, J.-P.: Cohomological invariants in Galois cohomology. In: University Lecture Series, vol. 28. Am. Math. Soc. (2003)

  7. Grothendieck, A.: Éléments de géométrie algébrique IV, étude locale des schémas et des morphismes de schémas, première partie. In: Publ. Math. Inst. Hautes Études Sci., vol. 20. Bures-sur-Yvette (1964)

  8. Grothendieck, A.: Le groupe de Brauer: I. Algèbres d’Azumaya et interprétations diverses. Séminaire Bourbaki 9, 199–219 (1964–1966) (Exposé No. 290)

    Google Scholar 

  9. Gille, P., Szamuely, T.: Central simple algebras and Galois cohomology. In: Cambridge Studies in Advanced Mathematics, vol. 101. Cambridge University Press, Cambridge (2006)

    Google Scholar 

  10. Kahn, B.: Cohomological approaches to SK 1 and SK 2 of central simple algebras. Doc. Math. Extra Volume: Andrei A. Suslin’s Sixtieth Birthday, 317–369 http://www.math.uiuc.edu/documenta/vol-suslin/vol-suslin-eng.html (2010)

  11. Knus, M.-A., Merkurjev, A., Rost, M., Tignol, J.-P.: The book of involutions. Am. Math. Soc. Colloq. Publ., vol. 44 (1998)

  12. Matsumura, H.: Commutative ring theory. Cambridge Studies in Advanced Mathematics, vol. 8. Cambridge University Press, Cambridge (1986) (Translated from the Japanese by M. Reid)

    MATH  Google Scholar 

  13. Merkurjev, A.: K-theory of simple algebras. In: K-theory and Algebraic Geometry: Connections with Quadratic Forms and Division Algebras (Santa Barbara, CA, 1992). In: Proceedings of Symposia in Pure Mathematics, vol. 58, pp. 65–83. Amer. Math. Soc., Providence, RI (1995)

  14. Merkurjev, A.: Invariants of algebraic groups. J. Reine Angew. Math. 508, 127–156 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  15. Merkurjev, A.: The group SK 1 for simple algebras. K-Theory 37(3), 311–319 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Milnor, J.: Algebraic K-theory and quadratic forms. Invent. Math. 9, 318–344 (1969/1970)

    Article  MathSciNet  Google Scholar 

  17. Nakayama, T., Matsushima, Y.: Über die multiplikative Gruppe einer p-adischen divisionsalgebra. Proc. Imp. Acad. Tokyo 19, 622–628 (1943)

    Article  MathSciNet  MATH  Google Scholar 

  18. Orlov, D., Vishik, A., Voevodsky, V.: An exact sequence for \(K^{M}_{\ast}/2\) with applications to quadratic forms. Ann. Math. 165(1), 1–13 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Platonov, V.: The Tannaka–Artin problem and reduced K-theory. Math. USSR Izv. 10(2), 211–243 (1976) (English translation)

    Article  Google Scholar 

  20. Prokopchuk, A., Tikhonov, S., Yanchevskiĭ, V.: Об обших элементах в группах SK1 для центральных простых алгебр. Vestsī Nats. Akad. Navuk Belarusī Ser. Fīz.-Mat. Navuk 126(3):35–42 (2008)

    Google Scholar 

  21. Rehmann, U., Tikhonov, S., Yanchevskiĭ, V.: Symbols and cyclicity of algebras after a scalar extension. Fundam. Prikl. Mat. 14(6), 193–209 (2008)

    Google Scholar 

  22. Rost, M.: Chow groups with coefficients. Doc. Math. J. DMV 1, 319–393 (1996)

    MathSciNet  MATH  Google Scholar 

  23. Rost, M., Serre, J.-P., Tignol, J.-P.: La forme trace d’une algèbre simple centrale de degré 4. C. R. Math. Acad. Sci. Paris 342(2), 83–87 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Serre, J.-P.: Galois Cohomology. Springer Monographs in Mathematics. Springer, Berlin (2002)

    Google Scholar 

  25. Suslin, A.: SK 1 of division algebras and Galois cohomology. In: Algebraic K-theory. Adv. Soviet Math., vol. 4, pp. 75–99. Amer. Math. Soc., Providence, RI (1991)

  26. Suslin, A.: SK 1 of division algebras and Galois cohomology revisited. In: Proceedings of the St. Petersburg Mathematical Society, vol. XII. Amer. Math. Soc. Transl. Ser. 2, vol. 219, pp. 125–147. Amer. Math. Soc., Providence, RI (2006)

  27. Wadsworth, A.: Valuation theory on finite dimensional division algebras. In: Valuation Theory and its Applications, vol. I (Saskatoon, SK, 1999). Fields Inst. Commun., vol. 32, pp. 385–449. Amer. Math. Soc., Providence, RI (2002)

  28. Wadsworth, A.: Unitary SK 1 of semiramified graded and valued division algebras (2010). http://arxiv.org/abs/1009.3904

  29. Wang, S.: On the commutator group of a simple algebra. Am. J. Math. 72, 323–334 (1950)

    Article  MATH  Google Scholar 

  30. Weibel, C.: An introduction to homological algebra. In: Cambridge Studies in Advanced Mathematics, vol. 38. Cambridge University Press, Cambridge (1997)

    Google Scholar 

  31. Wouters, T.: Cohomological invariants of SK 1. Ph.D. thesis, K.U.Leuven (2010)

  32. Wouters, T.: L’invariant de Suslin en caractéristique positive. J. K-Theory 5, 559–605 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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  1. Departement Wiskunde, K.U.Leuven, Celestijnenlaan 200B bus 2400, 3001, Leuven, Belgium

    Tim Wouters

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Wouters, T. Comparing Invariants of SK1 . Algebr Represent Theor 16, 729–745 (2013). https://doi.org/10.1007/s10468-011-9327-x

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