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Generalized periodicity theorems

Leonid Positselski

TL;DR

The paper develops a unifying framework for periodicity theorems in homological algebra via cotorsion pairs generated by a class S of FP${}_ ext{∞}$ modules. It defines A = {}^{⊥1}B, B = S^{⊥1}, C = varinjlim S, and D = C^{⊥1}, then proves two core periodicity statements: (a) any A-periodic object in C actually lies in A, and (b) any D-periodic object in B lies in D. These results generalize flat/projective, fp-projective, fp-injective/injective, and cotorsion periodicity and extend to Grothendieck categories and exact categories, with a two-part development culminating in a broad Theorem A for κ-presentable and deconstructible settings. The framework leverages deconstructibility, pure exact structures, and Eklof–Trlifaj theory in efficient Grothendieck-type exact categories to transfer periodicity phenomena across direct-limit closures and transfinitely iterated extensions. Overall, the work provides a cohesive, category-theoretic approach to periodicity that subsumes and extends classical module-theoretic results, enabling applications to a wide range of abelian and exact 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.

Generalized periodicity theorems

TL;DR

The paper develops a unifying framework for periodicity theorems in homological algebra via cotorsion pairs generated by a class S of FP modules. It defines A = {}^{⊥1}B, B = S^{⊥1}, C = varinjlim S, and D = C^{⊥1}, then proves two core periodicity statements: (a) any A-periodic object in C actually lies in A, and (b) any D-periodic object in B lies in D. These results generalize flat/projective, fp-projective, fp-injective/injective, and cotorsion periodicity and extend to Grothendieck categories and exact categories, with a two-part development culminating in a broad Theorem A for κ-presentable and deconstructible settings. The framework leverages deconstructibility, pure exact structures, and Eklof–Trlifaj theory in efficient Grothendieck-type exact categories to transfer periodicity phenomena across direct-limit closures and transfinitely iterated extensions. Overall, the work provides a cohesive, category-theoretic approach to periodicity that subsumes and extends classical module-theoretic results, enabling applications to a wide range of abelian and exact categories.

Abstract

Let be a ring and be a class of strongly finitely presented (FP) -modules closed under extensions, direct summands, and syzygies. Let be the (hereditary complete) cotorsion pair generated by in , and let be the (also hereditary complete) cotorsion pair in which . We show that any -periodic module in belongs to , and any -periodic module in belongs to . 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.
Paper Structure (7 sections, 41 theorems, 14 equations)

This paper contains 7 sections, 41 theorems, 14 equations.

Table of Contents

  1. Generalized Fp-injective/Injective and Cotorsion Periodicity
  2. Generalized Flat/Projective and Fp-projective Periodicity I
  3. Direct Limit Closures of Classes of Finitely Presentables
  4. Exact Categories of Grothendieck Type
  5. Pure Exact Structure and Deconstructibility
  6. Classes of $\kappa$-Presentables and Their $\kappa$-Direct Limit Closures
  7. Generalized Flat/Projective and Fp-projective Periodicity II

Key Result

Proposition 1.1

Let $\mathsf K$ be an abelian category and $\mathsf L\subset\mathsf L'\subset\mathsf M$ be three classes of objects in $\mathsf K$. Consider the following two properties: In this setting, the implication (1) $\Longrightarrow$ (2) holds true. If countable coproducts exist and are exact in $\mathsf K$, the classes $\mathsf L$ and $\mathsf M$ are closed under countable coproducts in $\mathsf K$, and

Theorems & Definitions (93)

  • Proposition 1.1
  • proof
  • proof : Proof of Theorem 0(b) from Section \ref{['introd-theorem-zero']}
  • Theorem 1.2
  • proof
  • proof : Proof of Theorem B from Section \ref{['introd-theorem-B']}
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 83 more