Periodic resolution of a finite cyclic group
= Periodic resolution of a finite cyclic group
If $C_m=\langle t\rangle$ and $N=1+t+\cdots+t^{m-1}$, a <free resolution> of the trivial $\mathbb ZC_m$-module alternates multiplication by $t-1$ and $N$. Applying $\operatorname{Hom}_{\mathbb ZC_m}(-,\mathbb Z)$ for the trivial action alternates the zero map and multiplication by $m$.