Krull dimension of the negligible quotient in mod p cohomology of a finite group

Abstract

For a finite group G, a G-module M, and a field F, an element u∈Hd(G,M) is negligible over F if for each field extension L/F and every continuous group homomorphism from Gal(Lsep/L) to G, u belongs to the kernel of the induced homomorphism Hd(G,M)→Hd(L,M). For p a prime and a trivial G-action on the coefficients, the negligible elements in the cohomology ring H⁎(G,Z/pZ) form an ideal. We compute the generators of the negligible ideal in the mod p cohomology of elementary abelian p-groups. We further show that when p is odd or p=2 and either |G| is odd or F is not formally real, the Krull dimension of the quotient of mod p cohomology by the negligible ideal is 0. However, when p=2, |G| is even, and F is formally real, the Krull dimension of the quotient of mod 2 cohomology of a finite 2-group by the negligible ideal is 1.