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arXiv · 2301.00708

Generalized periodicity theorems

Leonid Positselski
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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.

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Leonid Positselski. 2023-01-02. Generalized periodicity theorems. https://doi.org/10.1016/j.jpaa.2025.107962

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