Generalized periodicity theorems

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 $\text{Mod-}R$, and let $(\mathsf{C},\mathsf{D})$ be the (also hereditary complete) cotorsion pair in which $\mathsf{C}=\underrightarrow{\lim }\mathsf{A}=\underrightarrow{\lim }\mathsf{S}$. 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.

LaTeX 2e with with xy-pic; 42 pages, 3 commutative diagrams; v.5: new Propositions 5.2 and 6.2 inserted, Propositions 3.1 and 6.1 made more general, Propositions 5.3, 6.3 and 6.4 (former 5.2, 6.2 and 6.5) rewritten; v.6: the proof of Proposition 6.2 spelled out in more detail; v.7: several misprints corrected

References in corpus (15)

• On exact categories and applications to triangulated adjoints and model structures
• Singular compactness and definability for $\Sigma$-cotorsion and Gorenstein modules
• Deconstructibility and the Hill lemma in Grothendieck categories
• Exact model categories, approximation theory, and cohomology of quasi-coherent sheaves
• On compactly generated torsion pairs and the classification of co-t-structures for commutative noetherian rings
• Periodic modules and acyclic complexes
• The tilting-cotilting correspondence
• Cotilting Sheaves on Noetherian Schemes
• Fp-projective periodicity
• Projective covers of flat contramodules
• Closure properties of $\underrightarrow{\lim }\mathcal{C}$
• Coderived and contraderived categories of locally presentable abelian DG-categories
• Flat quasi-coherent sheaves as directed colimits, and quasi-coherent cotorsion periodicity
• Local, colocal, and antilocal properties of modules and complexes over commutative rings
• Locally coherent exact categories

Cited by in corpus (1)

• Contraderived categories of CDG-modules