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

For a finite group G, a G-module M, and a field F, an element u ∈ H d ( G , M ) is negligible over F if for each field extension L / F and every continuous group homomorphism from Gal ( L sep / L ) to G, u belongs to the kernel of the induced homomorphism H d ( G , M ) → H d ( L , M ) . For p a prime and a trivial G-action on the coefficients, the negligible elements in the cohomology ring H ⁎ ( G , Z / p Z ) 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.

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Alexander Merkurjev reports financial support was provided by National Science Foundation.