# Periodic modules and acyclic complexes

## TL;DR

This paper investigates the properties of $ ext{C}$-periodic modules and their implications for acyclic complexes and Gorenstein homological algebra, generalizing previous results and establishing new connections in the category of chain complexes.

## Contribution

It introduces a general framework for $ ext{C}$-periodic modules, extends known results, and applies these to prove that complexes of cotorsion modules are dg-cotorsion.

## Key findings

- Acyclic complexes of cotorsion modules have cotorsion cycles.
- Every map from an acyclic flat complex to a cotorsion complex is null-homotopic.
- Every complex of cotorsion modules is dg-cotorsion.

## Abstract

We study the behaviour of modules $M$ that fit into a short exact sequence $0\to M\to C\to M\to 0$, where $C$ belongs to a class of modules $\mathcal C$, the so-called $\mathcal C$-periodic modules. We find a rather general framework to improve and generalize some well-known results of Benson and Goodearl and Simson. In the second part we will combine techniques of hereditary cotorsion pairs and presentation of direct limits, to conclude, among other applications, that if $M$ is any module and $C$ is cotorsion, then $M$ will be also cotorsion. This will lead to some meaningful consequences in the category $\textrm{Ch}(R)$ of unbounded chain complexes and in Gorenstein homological algebra. For example we show that every acyclic complex of cotorsion modules has cotorsion cycles, and more generally, every map $F\to C$ where $C$ is a complex of cotorsion modules and $F$ is an acyclic complex of flat cycles, is null-homotopic. In other words, every complex of cotorsion modules is dg-cotorsion.