Research proof index

Let M be a representation of a finite cyclic group G. Suppose there are even and positive integers e and o with e even and o odd, such that Hᵉ(G,M) and Hᵒ(G,M) are both zero. Then Hⁿ(G,M) is zero for all n > 0.

Let M be a representation of a finite cyclic group G. Then the following square commutes coind₁'.obj M -------> coind₁'.obj M | | | | ↓ ↓ ind₁'.obj M -------> ind₁'.obj M The vertical maps are the canonical isomorphism ind₁'_iso_coind₁ and the horizontal maps are map₁ and map₂.

If σ generates H²(G,M) then the map H²(G,M) ⟶ H²(G,split σ) is zero.

For any subgroup H of G, the connecting hommorphism in the splitting module long exact sequence H¹(H,aug) ⟶ H²(H,M) is an isomorphism.

Given a 2-cocycle σ, the image of σ in the splitting module of σ is equal to the coboundary of τ σ.

To check that a finite group has trivial Tate cohomology, it's enough to show it has trivial cohomology and trivial homology, and that the 0-th and -1st Tate cohomology groups are trivial.