Integral cohomology of a finite cyclic group

For the trivial action of the finite cyclic group $C_m$ on $\mathbb{Z}$,for $k\ge 1$. The periodic resolution of a finite cyclic group proves this directly.

If $C_m=⟨t⟩$ and $N=1+t+\cdots +t^{m-1}$, a free resolution of the trivial $\mathbb{Z}{C}_m$-module alternates multiplication by $t-1$ and $N$. Applying ${\mathrm{Hom}}_{\mathbb{Z}{C}_m}(-,\mathbb{Z})$ for the trivial action alternates the zero map and multiplication by $m$.