Dimension shifting in group cohomology ($H^r(G,M^{\prime })\simeq H^{r+1}(G,M)$)

Embed a module $M$ over the group ring $\mathbb{Z}[G]$ in a module $I$ with vanishing positive group cohomology, and write $M^{\prime }=I/M$. The connecting map in the long exact group cohomology sequence of $0\to M\to I\to M^{\prime }\to 0$ is an isomorphism $H^r(G,M^{\prime })\to H^{r+1}(G,M)$ for $r\ge 1$, since the groups involving $I$ on both sides vanish. For a finite group and a module $I$ with vanishing Tate cohomology of a finite group in every degree, the same argument shifts every Tate degree. Repeating this construction transports computations to a convenient resolution.