Primitives and central detection numbers in group cohomology

• Autores: Nicholas Kuhn
• Localización: Advances in mathematics, ISSN 0001-8708, Vol. 216, Nº 1, 2007, págs. 387-442
• Idioma: inglés
• DOI: 10.1016/j.aim.2007.05.015
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• Resumen

  • Abstract Fix a prime p. Given a finite group G, let H*(G) denote its mod p cohomology. In the early 1990s, Henn, Lannes, and Schwartz introduced two invariants d0(G) and d1(G) of H*(G) viewed as a module over the mod p Steenrod algebra. They showed that, in a precise sense, H*(G) is respectively detected and determined by Hd(CG(V)) for dd0(G) and dd1(G), with V running through the elementary abelian p-subgroups of G.

    The main goal of this paper is to study how to calculate these invariants. We find that a critical role is played by the image of the restriction of H*(G) to H*(C), where C is the maximal central elementary abelian p-subgroup of G. A measure of this is the top degree e(G) of the finite dimensional Hopf algebra , a number that tends to be quite easy to calculate.

    Our results are complete when G has a p-Sylow subgroup P in which every element of order p is central. Using the Benson¿Carlson duality, we show that in this case, d0(G)=d0(P)=e(P), and a similar exact formula holds for d1. As a bonus, we learn that He(G)(P) contains nontrivial essential cohomology, reproving and sharpening a theorem of Adem and Karagueuzian.

    In general, we are able to show that d0(G)max{e(CG(V))|V

    En route we study two objects of independent interest. If C is any central elementary abelian p-subgroup of G, then H*(G) is an H*(C)-comodule, and we prove that the subalgebra of H*(C)-primitives is always Noetherian of Krull dimension equal to the p-rank of G minus the p-rank of C. If the depth of H*(G) equals the rank of Z(G), we show that the depth essential cohomology of G is nonzero (reproving and extending a theorem of Green), and Cohen¿Macauley in a certain sense, and prove related structural results.