Generalized periodicity theorems

TL;DR

This paper generalizes periodicity theorems for modules over rings, unifying various classical results and extending their applicability to arbitrary rings and Grothendieck categories.

Contribution

It introduces a broad framework that unifies and extends flat, projective, fp-projective, fp-injective, and cotorsion periodicity theorems within a common setting.

Findings

Any $ ext{A}$-periodic module in $ ext{C}$ belongs to $ ext{A}$.

Any $ ext{D}$-periodic module in $ ext{B}$ belongs to $ ext{D}$.

Results apply to modules over arbitrary rings and Grothendieck categories.

Abstract

Let $R$ be a ring and $\mathsf S$ be a class of strongly finitely presented (FP${}_\infty$) $R$-modules closed under extensions, direct summands, and syzygies. Let $(\mathsf A,\mathsf B)$ be the (hereditary complete) cotorsion pair generated by $\mathsf S$ in $\textsf{Mod-}R$, and let $(\mathsf C,\mathsf D)$ be the (also hereditary complete) cotorsion pair in which $\mathsf C=\varinjlim\mathsf A=\varinjlim\mathsf S$. We show that any $\mathsf A$-periodic module in $\mathsf C$ belongs to $\mathsf A$, and any $\mathsf D$-periodic module in $\mathsf B$ belongs to $\mathsf D$. Further generalizations of both results are obtained, so that we get a common generalization of the flat/projective and fp-projective periodicity theorems, as well as a common generalization of the fp-injective/injective and cotorsion periodicity theorems. Both are applicable to modules over an arbitrary ring, and in fact, to Grothendieck categories.