Periodic modules and acyclic complexes

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 $\text{Ch}(R)$ 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.

Cited by in corpus (12)

• Singular compactness and definability for $\Sigma$-cotorsion and Gorenstein modules
• Fp-projective periodicity
• Flat comodules and contramodules as directed colimits, and cotorsion periodicity
• Flat quasi-coherent sheaves as directed colimits, and quasi-coherent cotorsion periodicity
• Local, colocal, and antilocal properties of modules and complexes over commutative rings
• Locally coherent exact categories
• Homological theory of representations having pure acyclic injective resolutions
• Finiteness criteria for Gorenstein homological dimension and some invariants of groups
• Generalized periodicity theorems
• A contramodule generalization of Neeman's flat and projective module theorem
• A relative version of Bass' theorem about finite-dimensional algebras
• Contraderived categories of CDG-modules