Periodicity, for a module \(M\) over a ring \(R\), refers to the existence of a short exact sequence \(0 \rightarrow M \rightarrow A \rightarrow M \rightarrow 0\). If \(A\) belongs to a particular class \(\mathbf A\) of \(R\)-modules, then \(M\) is said to be \(\mathbf A\)-periodic. It is known that every flat \(\mathbf{Proj}\)-periodic module is projective [D. J. Benson and K. R. Goodearl, Pac. J. Math. 196, No. 1, 45–67 (2000; Zbl 1073.20500)], every fp-injective \(\mathbf{Inj}\)-periodic module is injective [J. Št’ovíček, “On purity and applications to coderived and singularity categories”, Preprint, arXiv:1412.1615], and every \(\mathbf{Cot}\)-periodic module is cotorsion [S. Bazzoni et al., Algebr. Represent. Theory 23, No. 5, 1861–1883 (2020; Zbl 1455.16006)], where \(\mathbf{Proj}\), \(\mathbf{Inj}\), and \(\mathbf{Cot}\) are the classes of projective, injective, and cotorsion \(R\)-modules, respectively. If a pure exact sequence \(0 \rightarrow M \rightarrow A \rightarrow M \rightarrow 0\) with \(A \in \mathbf{A}\) exists, \(M\) is said to be pure-\(\mathbf A\)-periodic. It is also known that every pure-\(\mathbf{PProj}\)-periodic module is pure-projective [D. Simson, Pac. J. Math. 207, No. 1, 235–256 (2002; Zbl 1056.16004)], and every pure-\(\mathbf{PInj}\)-periodic module is pure-injective [Št’ovíček, loc. cit.], where \(\mathbf{PProj}\) and \(\mathbf{PInj}\) are the respective classes of pure-projective and pure-injective \(R\)-modules.
One of the main results of the present paper is that every \(\mathbf{FpProj}\)-periodic \(R\)-module is weakly fp-projective in a suitable sense, where \(\mathbf{FpProj}\) is the class of fp-projective \(R\)-modules. This extends a result of J. Šaroch and J. Št’ovíček [Sel. Math., New Ser. 26, No. 2, Paper No. 23, 40 p. (2020; Zbl 1444.16010)], who proved that when \(R\) is coherent, every \(\mathbf{FpProj}\)-periodic \(R\)-module is fp-projective. (The weakly fp-projective right \(R\)-modules are fp-projective if and only if \(R\) is right coherent.)
Periodicity for flat \(\mathbf{Proj}\)-periodic modules was rediscovered and strengthened by A. Neeman [Invent. Math. 174, No. 2, 255–308 (2008; Zbl 1184.18008)], who proved that every acyclic complex of projective \(R\)-modules whose modules of cocycles are flat must be contractible, and that any morphism from a complex of projective \(R\)-modules to an acyclic complex of flat \(R\)-modules whose modules of cocycles are flat must be homotopic to zero. This type of homological periodicity is also obtained here for fp-projectivity: For any acyclic complex of fp-projective \(R\)-modules, the modules of cocycles must be weakly fp-projective; and any morphism from a complex of fp-projective \(R\)-modules to an acyclic complex of fp-injective \(R\)-modules whose modules of cocycles are fp-injective must be homotopic to zero.
All the mentioned main results are proved not just for the category of modules over a ring but for any locally finitely presentable abelian category.
As an application of their periodicity results, the authors prove that if \(\mathbf K\) is a locally coherent abelian category and \(\mathbf A\) the full subcategory of fp-projective objects in \(\mathbf K\), endowed with the inherited exact category structure, then the inclusion \(\mathbf{A} \rightarrow \mathbf{K}\) induces an equivalence of the unbounded derived categories \(D({\mathbf A})\) and \(D({\mathbf K})\). The paper concludes with a number of counterexamples to a potential strengthening of Simson’s periodicity theorem. Namely, over various rings \(R\), there exist \(\mathbf{PProj}\)-periodic \(R\)-modules which are not pure-projective.