Integral cohomology of a finite cyclic group

For the trivial group action of the finite cyclic group $C_m$ on the integers,The periodic resolution of a finite cyclic group alternates multiplication by $t-1$ and the norm $N=1+t+\cdots +t^{m-1}$. Applying the Hom functor with trivial coefficients makes those maps zero and multiplication by $m$, respectively. Their kernels and images give the displayed group cohomology. In particular $H^1=0$, while $H^2$ detects the cyclic central extensions appearing in second group cohomology classifies group extensions.

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 $m$.