Mathematische Zeitschrift

, Volume 244, Issue 4, pp 711–723 | Cite as

On Carlson's depth conjecture in group cohomology



We establish a weak form of Carlson's conjecture on the depth of the mod-p cohomology ring of a p-group. In particular, Duflot's lower bound for the depth is tight if and only if the cohomology ring is not detected on a certain family of subgroups. The proofs use the structure of the cohomology ring as a comodule over the cohomology of the centre via the multiplication map. We demonstrate the existence of systems of parameters (so-called polarised systems) which are particularly well adapted to this comodule structure.


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  1. 1.
    Benson, D.J.: The image of the transfer map. Archiv. Math. 61, 7–11 (1993)MathSciNetMATHGoogle Scholar
  2. 2.
    Benson, D.J.: Polynomial invariants of finite groups. London Math. Soc. Lecture Note Series, vol. 190. Cambridge University Press, Cambridge, 1993Google Scholar
  3. 3.
    Broto, C., Henn, H.-W.: Some remarks on central elementary abelian p-subgroups and cohomology of classifying spaces. Quart. J. Math. Oxford Ser. 44(2), 155–163 (1993)MATHGoogle Scholar
  4. 4.
    Carlson, J.F.: Depth and transfer maps in the cohomology of groups. Math. Z. 218, 461–468 (1995)MathSciNetMATHGoogle Scholar
  5. 5.
    Carlson, J.F.: Problems in the calculation of group cohomology. In: P. Dräxler, G.O. Michler, C.M. Ringel, editors, Computational methods for representations of groups and algebras (Essen, 1997), pp. 107–120. Birkhäuser, Basel, 1999Google Scholar
  6. 6.
    Duflot, J.: Depth and equivariant cohomology. Comment. Math. Helv. 56, 627–637 (1981)MathSciNetMATHGoogle Scholar
  7. 7.
    Evens, L.: The cohomology of groups. Oxford Univ. Press, Oxford, 1991Google Scholar
  8. 8.
    Leary, I.J.: The integral cohomology rings of some p-groups. Math. Proc. Cambridge Philos. Soc. 110, 25–32 (1991)MathSciNetMATHGoogle Scholar
  9. 9.
    Milgram, R.J., Tezuka, M.: The geometry and cohomology of M 12. II. Bol. Soc. Mat. Mexicana 1(3), 91–108 (1995)MATHGoogle Scholar
  10. 10.
    Minh, P.A.: Essential cohomology and extraspecial p-groups. Trans. Amer. Math. Soc. 353, 1937–1957 (2000)Google Scholar
  11. 11.
    Quillen, D.: The mod-2 cohomology rings of extra-special 2-groups and the spinor groups. Math. Ann. 194, 197–212 (1971)MATHGoogle Scholar
  12. 12.
    Wilkerson, C.: A primer on the Dickson invariants. In: H.R. Miller, S.B. Priddy, editors, Proceedings of the Northwestern Homotopy Theory Conference (Evanston, Ill., 1982), Contemporary Math., vol. 19, p. 421–434. Amer. Math. Soc., Providence, RI, 1983Google Scholar

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© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  1. 1.Fachbereich 7 MathematikBergische Universität WuppertalWuppertalGermany

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