Negligible degree two cohomology of finite groups

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 group homomorphism Gal ( L sep / L ) → G , u belongs to the kernel of the induced homomorphism H d ( G , M ) → H d ( L , M ) . We determine the group of negligible elements in H 2 ( G , M ) for every abelian group M with trivial G-action.

对于一个有限群\(G\)、一个\(G\)-模\(M\)以及一个域\(F\)，如果对于每个域扩张\(L/F\)以及每一个群同态\(\mathrm{Gal}(L^{\mathrm{sep}}/L)\to G\)，元素\(u\in H^d(G,M)\)都属于诱导同态\(H^d(G,M)\to H^d(L,M)\)的核，那么\(u\)在\(F\)上是可忽略的。我们确定了对于每个具有平凡\(G\)-作用的阿贝尔群\(M\)，\(H^2(G,M)\)中可忽略元素的群。