A general construction of $n$-angulated categories using periodic injective resolutions

Let $\mathcal{C}$ be an additive category equipped with an automorphism $\mathrm{\Sigma }$. We show how to obtain $n$-angulations of $(\mathcal{C},\mathrm{\Sigma })$ using some particular periodic injective resolutions. We give necessary and sufficient conditions on $(\mathcal{C},\mathrm{\Sigma })$ admitting an $n$-angulation. Then we apply these characterizations to explain the standard construction of $n$-angulated categories and the $n$-angulated categories arising from some local rings. Moreover, we obtain a class of new examples of $n$-angulated categories from quasi-periodic selfinjective algebras.