Cyclic-extension shift of group-ring cohomology ($H^{j+1}(N⋊\mathbb{Z};\mathbb{Z}[N⋊\mathbb{Z}])\cong H^j(N;\mathbb{Z}N)$)

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 $\mathbb{Z}N$, 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 $H^d(N;\mathbb{Z}N)\ne 0$, so the cyclic extension has dimension $d+1$.