Periodic modules and acyclic complexes
Silvana Bazzoni, Manuel Cortés Izurdiaga, Sergio Estrada
TL;DR
The paper develops a general framework for $\mathcal{C}$-periodic modules, using locally split short exact sequences and countable generation to identify when such modules must be trivial, and then dualizes these ideas through hereditary cotorsion pairs to obtain broad consequences for cotorsion, Gorenstein modules, and acyclic complexes. It proves a key splitting theorem for sequences with locally split maps and countable-generation targets, yielding triviality results for pure periodic modules and corollaries for Gorenstein injectives. The results extend to chain complexes, showing that acyclic complexes of cotorsion modules have cotorsion cycles and that $dg$-cotorsion agrees with $dw$-cotorsion in this context, with further implications in finitely accessible categories. Overall, the work ties module-theoretic periodicity to derived-category and homological-algebra structures, informing both theoretical understanding and potential applications in Gorenstein homological algebra.
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.