= Cyclic-extension shift of group-ring cohomology
{title2=$H^{j+1}(N\rtimes\mathbb Z;\mathbb Z[N\rtimes\mathbb Z])\cong H^j(N;\mathbb ZN)$}
If $N$ has a finite resolution by finitely generated projective modules, restriction of the semidirect-product group ring gives a direct sum of copies of $\mathbb ZN$, indexed by the integers. On its kernel cohomology the cyclic generator shifts those copies, possibly twisting each copy by an automorphism. Invariants are zero because elements have finite support; coinvariants identify with a single copy. The two-column <Lyndon–Hochschild–Serre spectral sequence> therefore gives the displayed isomorphism of abelian groups. A group with such a finite resolution and cohomological dimension $d$ has $H^d(N;\mathbb ZN)\ne0$, so the cyclic extension has dimension $d+1$.
Back to article page
