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

TL;DR

This paper introduces a general method for constructing n-angulated categories using periodic injective resolutions, providing new insights and examples in the theory of higher homological algebra.

Contribution

It offers necessary and sufficient conditions for (C,Σ) to admit an n-angulation and applies these to standard and novel examples, including those from local rings and selfinjective algebras.

Findings

Characterization of (C,Σ) admitting n-angulations

Construction of new n-angulated categories from selfinjective algebras

Explanation of standard n-angulated categories from local rings

Abstract

Let $\mathcal{C}$ be an additive category equipped with an automorphism $\Sigma$. We show how to obtain $n$-angulations of $(\mathcal{C},\Sigma)$ using some particular periodic injective resolutions. We give necessary and sufficient conditions on $(\mathcal{C},\Sigma)$ 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.