Flat quasi-coherent sheaves as directed colimits, and quasi-coherent cotorsion periodicity

Abstract

We show that every flat quasi-coherent sheaf on a quasi-compact quasi-separated scheme is a directed colimit of locally countably presentable flat quasi-coherent sheaves. More generally, the same assertion holds for any countably quasi-compact, countably quasi-separated scheme. Moreover, for three categories of complexes of flat quasi-coherent sheaves, we show that all complexes in the category can be obtained as directed colimits of complexes of locally countably presentable flat quasi-coherent sheaves from the same category. In particular, on a quasi-compact semi-separated scheme, every flat quasi-coherent sheaf is a directed colimit of flat quasi-coherent sheaves of finite projective dimension. In the second part of the paper, we discuss cotorsion periodicity in category-theoretic context, generalizing an argument of Bazzoni, Cortes-Izurdiaga, and Estrada. As the main application, we deduce the assertion that any cotorsion-periodic quasi-coherent sheaf on a quasi-compact semi-separated scheme is cotorsion.

LaTeX 2e with xy-pic; 31 pages, 2 commutative diagrams; v.2: Sections 1-2 completely rewritten and replaced with new Sections 1-4; v.3: new Remarks 2.3 and 3.6 inserted, the proof of Lemma 6.5 expanded, Remark 9.3 rewritten, references added and updated; v.4: several misprints corrected, references updated, the numbering of sections shifted to agree with the journal version

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• On exact categories and applications to triangulated adjoints and model structures
• Derived, coderived, and contraderived categories of locally presentable abelian categories
• Fp-projective periodicity
• Flat comodules and contramodules as directed colimits, and cotorsion periodicity
• Coderived and contraderived categories of locally presentable abelian DG-categories
• Locally coherent exact categories

Cited by in corpus (6)

• Roos axiom holds for quasi-coherent sheaves
• Generalized periodicity theorems
• A relative version of Bass' theorem about finite-dimensional algebras
• A contramodule generalization of Neeman's flat and projective module theorem
• Contraderived categories of CDG-modules
• The categories of corings and coalgebras over a ring are locally countably presentable