Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-26T00:55:05.488Z Has data issue: false hasContentIssue false
  • English
  • Français

The Nilpotent filtration and the action of automorphisms on the cohomology of finite p-groups

Published online by Cambridge University Press:  01 May 2008

NICHOLAS J. KUHN
NICHOLAS J. KUHN*
Affiliation:
Department of Mathematics, University of Virginia, Charlottesville, VA 22904, USA. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We study H*(P), the mod p cohomology of a finite p-group P, viewed as an –module. In particular, we study the conjecture, first considered by Martino and Priddy, that, if e ∈ is a nonzero idempotent, then the Krull dimension of eH*(P) equals the rank of P. We prove this for all p-groups when p is odd, and for many 2–groups.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 144 , Issue 3 , May 2008 , pp. 575 - 602
Copyright
Copyright © Cambridge Philosophical Society 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[Alp]Alperin, J. L.. Local representation theory, Camb. Stud. Adv. Math. 11 (Cambridge University Press, 1986).Google Scholar
[Asch]Aschbacher, M.. Finite group theory, 2nd edition. Camb. Stud. Adv. Math. 10 (Cambridge University Press, 2000).Google Scholar
[BoZ]Bourguiba, D. and Zarati, S.. Depth and the Steenrod algebra with an appendix by J. Lannes. Invent. Math. 128 (1997), 589–602.CrossRefGoogle Scholar
[BrH]Broto, C. and Henn, H.-W.. Some remarks on central elementar abelian p–subgroups and cohomology of classifying spaces, Quart. J. Math. 44 (1993), 155–163.CrossRefGoogle Scholar
[BrZ1]Broto, C. and Zarati, S.. Nil–localization of unstable algebras over the Steenrod algebra. Math. Zeit. 199 (1988), 525–537.CrossRefGoogle Scholar
[BrZ2]Broto, C. and Zarati, S.. On sub– Ap*–algebras of H*(V). Springer Ledre Note Math. 1509 (1992), 35–49.Google Scholar
[Ca]Carlson, J.. Mod 2 cohomology of 2 groups, MAGMA computer computations on the website http://www.math.uga.edu/simlvalero/cohointro.html.Google Scholar
[CTVZ]Carlson, J. F., Townsley, L., Valeri-Elizondo, L., and Zhang, M.. Cohomology rings of finite groups. With an appendix, Calculations of cohomology rings of groups of order dividing 64, by Carlson, Valeri-Elizondo and Zhang. Algebras and Applications 3 (Kluwer, D., 2003).Google Scholar
[Cr]Crabb, M. C.. Dickson-Mui invariants. Bull. London Math. Soc. 37 (2005), 846–856.CrossRefGoogle Scholar
[D]Dickson, L. E.. A fundamental system of invariants of the general modular linear group with a solution of the form problem. Trans. Amer. Math. Soc. 12 (1911), 75–98.CrossRefGoogle Scholar
[DS]Diethelm, T. and Stammbach, U.. On the module structure of the mod p cohomology of a p-group. Arch. Math. 43 (1984), 488–492.CrossRefGoogle Scholar
[D]Duflot, J.. Depth and equivariant cohomology. Comm. Math. Helv. 56 (1981), 627–637.CrossRefGoogle Scholar
[Gor]Gorenstein, D.. Finite Groups, 2nd edition. (Chelsea Publishing 1980).Google Scholar
[Gr]Green, D. J.. The essential ideal in group cohomology does not square to zero. J. Pure Appl. Al. 193 (2004), no. 1–3, 129–139.CrossRefGoogle Scholar
[HK]Harris, J. C. and Kuhn, N. J.. Stable decompositions of classifying spaces of finite abelian p-groups. Math. Proc. Camb. Phil. Soc. 103 (1988), 427–449.CrossRefGoogle Scholar
[H]Henn, H.-W.. Finiteness properties of injective resolutions of certain unstable modules over the Steenrod algebra and applications. Math. Ann. 291 (1991), 191–203.CrossRefGoogle Scholar
[HLS1]Henn, H.-W., Lannes, J. and Schwartz, L.. The categories of unstable modules and unstable algebras modulo nilpotent objects. Amer. J. Math. 115 (1993), 1053–1106.CrossRefGoogle Scholar
[HLS2]Henn, H.-W., Lannes, J., and Schwartz, L.. Localizations of unstable A-modules and equivariant mod p cohomology. Math. Ann. 301 (1995), 23–68.CrossRefGoogle Scholar
[HP]Henn, H.-W. and Priddy, S.. p–nilpotence, classifying space indecomposability, and other properties of almost all finite groups. Comm. Math. Helv. 69 (1994), 335–350.CrossRefGoogle Scholar
[Hi]Higginbottom, R.. Ph.D. thesis (University of Virginia, 2005).Google Scholar
[K1]Kuhn, N. J.. Character rings in algebraic topology. Advances in Homotopy Theory (Cortona 1988), London Math. Soc. Lectures Notes 139 (1989), 111–126.Google Scholar
[K2]Kuhn, N. J.. Generic representations of the finite general linear groups and the Steenrod algebra: I. Amer. J. Math. 116 (1994), 327–360.CrossRefGoogle Scholar
[K3]Kuhn, N. J.. On topologically realizing modules over the Steenrod algebra. Ann. Math. 141 (1995), 321–347.CrossRefGoogle Scholar
[K4]Kuhn, N. J.. Primitives and central detection numbers in group cohomology. Adv. Math. 216 (2007), 387–442.CrossRefGoogle Scholar
[LZ]Lannes, J., and Zarati, S.. Sur les mycal U–injectifs. Ann. Sci. Ec. Norm. Sup. 19 (1986), 303–333.CrossRefGoogle Scholar
[MP]Martino, J. and Priddy, S.. On the dimension theory of dominant summands, Adams Memorial Symposium on Algebraic Topology, 1 (Manchester, 1990). London Math. Soc. Lect. Note Ser. 175 (1992), 281–292.Google Scholar
[Mat]Matsumura, H.. Commutative algebra, 2nd edition. Math. Lect. Note. Series, (Benjamin, 1980).Google Scholar
[N]Nishida, G.. Stable homotopy type of classifying spaces of finite groups. Algebraic and topological theories (Kinosaki 1984) (Kinokuniya, Tokyo, 1986), 391–404.Google Scholar
[Q1]Quillen, D.. The spectrum of an equivariant cohomology ring I, Ann. Math. 94 (1971), 549–572.CrossRefGoogle Scholar
[Q2]Quillen, D.. The spectrum of an equivariant cohomology ring II. Ann. Math. 94 (1971), 573–602.CrossRefGoogle Scholar
[S1]Schwartz, L.. La filtration nilpotente de la catégorie U et la cohomologie des espaces de lacets. Algebraic Topology–Rational Homotopy (Louvain la Neuve, 1986), S. L. N. M. 1318 (1988), 208–218.CrossRefGoogle Scholar
[S2]Schwartz, L.. Modules over the Steenrod algebra and Sullivan's fixed point conjecture. Chicago Lectures in Math. (University Chicago Press, 1994).Google Scholar
[Sy]Symonds, P.. The action of automorphisms on the cohomology of a p-group. Math. Proc. Camb. Phil. Soc. 127 (1999), 495–496.CrossRefGoogle Scholar