Negligible degree two cohomology of finite groups

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 group homomorphism Gal(Lsep/L)→G, u belongs to the kernel of the induced homomorphism Hd(G,M)→Hd(L,M). We determine the group of negligible elements in H2(G,M) for every abelian group M with trivial G-action.