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Inventiones mathematicae

, Volume 172, Issue 3, pp 491–508 | Cite as

Essential dimension of finite p-groups

  • Nikita A. Karpenko
  • Alexander S. Merkurjev
Article

Abstract

We prove that the essential dimension and p-dimension of a p-group G over a field F containing a primitive p-th root of unity is equal to the least dimension of a faithful representation of G over F.

Keywords

Algebraic Group Isomorphism Class Essential Dimension Valuation Ring Faithful Representation 
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References

  1. 1.
    Artin, M.: Brauer–Severi varieties (Notes by A. Verschoren). In: van Oystaeyen, F.M.J., Verschoren, A.H.M.J. (eds.) Brauer Groups in Ring Theory and Algebraic Geometry (Wilrijk, 1981). Lect. Notes Math., vol. 917, pp. 194–210. Springer, Berlin (1982)CrossRefGoogle Scholar
  2. 2.
    Berhuy, G., Favi, G.: Essential dimension: a functorial point of view (after A. Merkurjev). Doc. Math. 8, 279–330 (2003) (electronic)Google Scholar
  3. 3.
    Berhuy, G., Reichstein, Z.: On the notion of canonical dimension for algebraic groups. Adv. Math. 198(1), 128–171 (2005)CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    Brosnan, P., Reichstein, Z., Vistoli, A.: Essential dimension and algebraic stacks. LAGRS preprint server, http://www.math.uni-bielefeld.de/lag/ (2007)Google Scholar
  5. 5.
    Buhler, J., Reichstein, Z.: On the essential dimension of a finite group. Compos. Math. 106(2), 159–179 (1997)CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Florence, M.: On the essential dimension of cyclic p-groups. Invent. Math. 171, 175–189 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Giraud, J.: Cohomologie non abélienne. Grundlehren Math. Wiss., vol. 179. Springer, Berlin (1971)zbMATHGoogle Scholar
  8. 8.
    Karpenko, N.A.: Grothendieck Chow motives of Severi–Brauer varieties (Russian). Algebra Anal. 7(4), 196–213 (1995) (transl. in St. Petersbg. Math. J. 7(4), 649–661 (1996))MathSciNetzbMATHGoogle Scholar
  9. 9.
    Karpenko, N.A.: On anisotropy of orthogonal involutions. J. Ramanujan Math. Soc. 15(1), 1–22 (2000)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Karpenko, N.A., Merkurjev, A.S.: Essential dimension of quadrics. Invent. Math. 153(2), 361–372 (2003)CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Karpenko, N.A., Merkurjev, A.S.: Canonical p-dimension of algebraic groups. Adv. Math. 205(2), 410–433 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Merkurjev, A.S.: Maximal indices of Tits algebras. Doc. Math. 1(12), 229–243 (1996) (electronic)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Milne, J.S.: Étale Cohomology. Princeton University Press, Princeton, N.J. (1980)zbMATHGoogle Scholar
  14. 14.
    Quillen, D.: Higher Algebraic K-Theory, I. Lect. Notes Math., vol. 341, pp. 85–147. Springer, Berlin (1973)Google Scholar
  15. 15.
    Reichstein, Z., Youssin, B.: Essential dimensions of algebraic groups and a resolution theorem for G-varieties. Canad. J. Math. 52(5), 1018–1056 (2000) (With an appendix by J. Kollár and E. Szabó)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Rowen, L.H.: Ring Theory, vol. II. Pure Appl. Math., vol. 128. Academic Press, Boston, MA (1988)Google Scholar
  17. 17.
    Serre, J.-P.: Linear Representations of Finite Groups. Grad. Texts Math., vol. 42. Springer, New York (1977) (Transl. from the 2nd French edn. by L.L. Scott)Google Scholar
  18. 18.
    Thomason, R.W.: Algebraic K-theory of group scheme actions. In: Algebraic Topology and Algebraic K-theory (Princeton, N.J., 1983). Ann. Math. Stud., vol. 113, pp. 539–563. Princeton Univ. Press, Princeton, NJ (1987)Google Scholar
  19. 19.
    Vistoli, A.: Intersection theory on algebraic stacks and on their moduli spaces. Invent. Math. 97(3), 613–670 (1989)CrossRefMathSciNetzbMATHGoogle Scholar
  20. 20.
    Zariski, O., Samuel, P.: Commutative Algebra, vol. II. Grad. Texts Math. vol. 29. Springer, New York (1975) (Reprint of the 1960 edn.)Google Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Institut de Mathématiques de JussieuUniversité Pierre et Marie Curie (Paris 6)Paris Cedex 05France
  2. 2.Department of MathematicsUniversity of CaliforniaLos AngelesUSA

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