Periodic resolution of a finite cyclic group

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 $m$.