= Dimension shifting in group cohomology
{title2=$H^r(G,M')\simeq H^{r+1}(G,M)$}
Embed a <module> $M$ over the <group ring> $\mathbb Z[G]$ in a module $I$ with vanishing positive <group cohomology>, and write $M'=I/M$. The connecting map in the long exact <group cohomology> sequence of $0\to M\to I\to M'\to0$ is an isomorphism $H^r(G,M')\to H^{r+1}(G,M)$ for $r\geq1$, 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.
Back to article page
