The Chern classes modulo pp of a regular representation

  • Bruno Kahn

    Institut de Mathematiques de Jussieu Universite Paris 7 Case 7012 75251 Paris Cedex 05 France
The Chern classes modulo $p$ of a regular representation cover
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Let GG be a finite group and ρ\rho a complex linear representation of GG. In 1961, Atiyah and Venkov independently defined Chern classes ci(ρ)c_i(\rho) with values in the integral or mod pp cohomology of GG. We consider here the mod pp Chern classes of the regular representation rGr_G of GG. Venkov claimed that ci(rG)=0c_i(r_G)=0 for i<pnpn1i<p^n-p^{n-1}, where pnp^n is the highest power of pp dividing G|G|; however his proof is only valid for GG elementary abelian. In this note, we show Venkov's assertion is valid for any GG. The proof also shows that the ci(rG)c_i(r_G) are pp-powers of cohomology classes invariant by Aut(G)Aut(G) as soon as GG is a non-abelian pp-group.

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Bruno Kahn, The Chern classes modulo pp of a regular representation. Doc. Math. 4 (1999), pp. 167–178

DOI 10.4171/DM/57