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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]
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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

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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