Advertisement

Manuscripta Mathematica

, Volume 139, Issue 3–4, pp 343–389 | Cite as

Unitary SK1 of semiramified graded and valued division algebras

  • A. R. WadsworthEmail author
Article

Abstract

We prove formulas for SK1(E, τ), which is the unitary SK1 for a graded division algebra E finite-dimensional and semiramified over its center T with respect to a unitary involution τ on E. Every such formula yields a corresponding formula for SK1(D, ρ) where D is a division algebra tame and semiramified over a Henselian valued field and ρ is a unitary involution on D. For example, it is shown that if \({\sf{E} \sim \sf{I}_0 \otimes_{\sf{T}_0}\sf{N}}\) where I 0 is a central simple T 0-algebra split by N 0 and N is decomposably semiramified with \({\sf{N}_0 \cong L_1\otimes_{\sf{T}_0} L_2}\) with L 1, L 2 fields each cyclic Galois over T 0, then
$${\rm SK}_1(\sf{E}, \tau) \,\cong\ {\rm Br}(({L_1}\otimes_{\sf{T}_0} {L_2})/\sf{T}_0;\sf{T}_0^\tau)\big/ \left[{\rm Br}({L_1}/\sf{T}_0;\sf{T}_0^\tau)\cdot {\rm Br}({L_2}/\sf{T}_0;\sf{T}_0^\tau) \cdot \langle[\sf{I}_0]\rangle\right].$$

Mathematics Subject Classification (2000)

16K20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Amitsur S.A., Rowen L.H., Tignol J.-P.: Division algebras of degree 4 and 8 with involution. Israel J. Math. 33, 133–148 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Amitsur S.A., Saltman D.J.: Generic abelian crossed products and p-algebras. J. Algebra 51, 76–87 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Arason J., Elman R.: Nilpotence in the Witt ring. Am. J. Math. 113, 861–875 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Chernousov V., Merkurjev A.: R-equivalence and special unitary groups. J. Algebra 209, 175–198 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Draxl P.: SK 1 von Algebren über vollständig diskret bewerteten Körpern und Galoiskohomologie abelscher Körpererweiterungen. J. Reine Angew. Math. 293/294, 116–142 (1977)MathSciNetGoogle Scholar
  6. 6.
    Ershov, Y.L.: Valuations of division algebras, and the group SK 1. Dokl. Akad. Nauk SSSR 239, 768–771 (1978) (in Russian); English transl., Soviet Math. Doklady 19, 395–399 (1978)Google Scholar
  7. 7.
    Ershov, Y.L.: Henselian valuations of division rings and the group SK 1. Mat. Sb. (N.S.) 117, 60–68 (1982) (in Russian); English transl., Math USSR-Sbornik 45, 63–71 (1983)Google Scholar
  8. 8.
    Gille, P.: Le problème de Kneser-Tits, Séminaire Bourbaki, Exp. No. 983, vol. 2007/2008. Astérisque 326, 39–81 (2010)Google Scholar
  9. 9.
    Haile D.E., Knus M.-A., Rost M., Tignol J.-P.: Algebras of odd degree with involution, trace forms and dihedral extensions. Israel J. Math. 96(part B), 299–340 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Hazrat R., Wadsworth A.R.: SK1 of graded division algebras. Israel J. Math. 183, 117–163 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Hazrat, R., Wadsworth, A.R.: Unitary SK1 of graded and valued division algebras. Proc. Lond. Math. Soc., to appear, preprint available at arXiv: 0911.3628Google Scholar
  12. 12.
    Hwang Y.-S., Wadsworth A.R.: Algebraic extensions of graded and valued fields. Commun. Algebra 27, 821–840 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Hwang Y.-S., Wadsworth A.R.: Correspondences between valued division algebras and graded division algebras. J. Algebra 220, 73–114 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Jacob B., Wadsworth A.: Division algebras over Henselian fields. J. Algebra 128, 126–179 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Kahn, B.: Cohomological approaches to SK1 and SK 2 of central simple algebras. Documenta Math., Extra Volume, Andrei A. Suslin’s Sixtieth Birthday, pp. 371–392 (electronic) (2010)Google Scholar
  16. 16.
    Karpenko N.A.: Codimension 2 cycles on Severi-Brauer varieties. K-Theory 13, 305–330 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Knus, M.-A., Merkurjev, A., Rost, M., Tignol, J.-P.: The Book of Involutions, AMS Coll. Pub., vol. 44. American Mathematical Society, Providence (1998)Google Scholar
  18. 18.
    Merkurjev A.S.: Generic element in SK 1 for simple algebras. K-Theory 7, 1–3 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Merkurjev A.S.: Invariants of algebraic groups. J. Reine Angew. Math. 508, 127–156 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Merkurjev A.S.: Cohomological invariants of simply connected groups of rank 3. J. Algebra 227, 614–632 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Merkurjev A.S.: The group SK 1 for simple algebras. K-Theory 37, 311–319 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Mounirh K.: Nondegenerate semiramified valued and graded division algebras. Commun. Algebra 36, 4386–4406 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Platonov, V.P.: On the Tannaka-Artin problem. Dokl. Akad. Nauk SSSR 221, 1038–1041 (1975) (in Russian); English transl., Soviet Math. Dokl. 16, 468–473 (1975)Google Scholar
  24. 24.
    Platonov, V.P.: The Tannaka-Artin problem and reduced K-theory. Izv. Akad. Nauk SSSR Ser. Mat. 40, 227–261 (1976) (in Russian); English transl., Math. USSR-Izvestiya 10, 211–243 (1976)Google Scholar
  25. 25.
    Platonov, V.P.: The reduced Whitehead group for cyclic algebras. Dokl. Akad. Nauk SSSR 228, 38–40 (1976) (in Russian); English transl., Soviet. Math. Doklady 17, 652–655 (1976)Google Scholar
  26. 26.
    Platonov, V.P.: The Infinitude of the reduced Whitehead group in the Tannaka-Artin Problem, Mat. Sb. 100(142), 191–200, 335 (1976) (in Russian); English transl., Math. USSR Sbornik 29, 167–176 (1976)Google Scholar
  27. 27.
    Platonov, V.P.: Birational properties of the reduced Whitehead group. Dokl. Akad. Nauk BSSR 21, 197–198, 283 (1977) (in Russian); English transl., pp. 7–9 in Selected Papers in K-theory. American Mathematical Society Translations, ser. 2, vol. 154. American Mathematical Society, Providence (1992)Google Scholar
  28. 28.
    Platonov V.P.: Algebraic groups and reduced K-theory. In: Leht O., (eds) Proceedings of the International Congress of Mathematicians (Helsinki 1978)., pp. 311–317. Acad. Sci. Fennica, Helsinki (1980)Google Scholar
  29. 29.
    Rehmann, U., Tikhonov, S.V., Yanchevskiĭ, V.I.: Symbols and cyclicity of algebras after a scalar extension. Fundam. Prikl. Mat., 14, 193–209 (2008) (in Russian); English transl., J. Math. Sci. (N. Y.) 164, 131–142 (2010)Google Scholar
  30. 30.
    Rowen L.H.: Central simple algebras. Israel J. Math. 29, 285–301 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Serre, J.-P: Local Fields, 2nd edn. Springer, New York (1995) (English trans. of Corps Locaux)Google Scholar
  32. 32.
    Suslin, A.A.: SK 1 of division algebras and Galois cohomology. In: Algebraic K-theory. Advances in Soviet Mathematics, Vol. 4, pp. 75–99. American Mathematical Society, Providence (1991)Google Scholar
  33. 33.
    Suslin, A.A.: SK 1 of division algebras and Galois cohomology revisited, In: Proc. St. Petersburg Math. Soc., Vol. XII, pp. 125–147. English transl., Amer. Math. Soc. Transl. Ser. 2, Vol. 219, Amer. Math. Soc., Providence (2006)Google Scholar
  34. 34.
    Tignol, J.-P.: Corps à à involution neutralisés par une extension abelienne elémentaire. In: Groupe de Brauer. Kervaire, M., Ojanguren, M. (eds.). Lecture Notes in Mathematics, No. 844, pp. 1–34. Springer, Berlin (1981)Google Scholar
  35. 35.
    Tignol J.-P.: Produits croisés abéliens. J. Algebra 70, 420–436 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Tignol J.-P., Wadsworth A.R.: Value functions and associated graded rings for semisimple algebras. Trans. Am. Math. Soc. 362, 687–726 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Tits, J.: Groupes de Whitehead de groupes algébriques simples sur un corps (d’après V. P. Platonov et al.). In: Séminaire Bourbaki, 29e année (1976/77), Exp. No. 505, Lecture Notes in Mathematics, vol. 677, pp. 218–236. Springer, Berlin (1978)Google Scholar
  38. 38.
    Voskresenskiĭ, V.E.: The reduced Whitehead group of a simple algebra. Uspehi Mat. Nauk 32, 247–248 (1977) (in Russian)Google Scholar
  39. 39.
    Voskresenskiĭ, V.E.: Algebraic Groups and Their Birational Invariants. Translations of Mathematical Monographs, vol. 179. American Mathematical Society, Providence (1998)Google Scholar
  40. 40.
    Wouters, T.: Comparing invariants of SK1, preprint, arXiv: 1003.1654v2Google Scholar
  41. 41.
    Yanchevskiĭ, V.I.: Simple algebras with involutions, and unitary groups. Mat. Sb. (N.S.) 93(135), 368–380, 487 (1974) (in Russian); English transl., Math. USSR-Sbornik 22, 372–385 (1974)Google Scholar
  42. 42.
    Yanchevskiĭ, V.I.: Reduced unitary K-theory. Dokl. Akad. Nauk SSSR 229, 1332–1334 (1976) (in Russian); English transl., Soviet Math. Dokl. 17, 1220–1223 (1976)Google Scholar
  43. 43.
    Yanchevskiĭ, V.I.: Reduced unitary K-Theory and division rings over discretely valued Hensel fields. Izv. Akad. Nauk SSSR Ser. Mat. 42, 879–918 (1978) (in Russian); English transl., Math. USSR Izvestiya 13, 175–213 (1979)Google Scholar
  44. 44.
    Yanchevskiĭ, V.I.: The inverse problem of reduced K-theory, Mat. Zametki 26, 475–482 (1979) (in Russian); English transl., A converse problem in reduced unitary K-theory. Math. Notes 26, 728–731 (1979)Google Scholar
  45. 45.
    Yanchevskiĭ, V.I.: Reduced unitary K-theory. Applications to algebraic groups. Mat. Sb. (N.S.) 110(152), 579–596 (1979) (in Russian); English transl., Math. USSR Sbornik 38, 533–548 (1981)Google Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California at San DiegoLa JollaUSA

Personalised recommendations