Flat comodules and contramodules as directed colimits, and cotorsion periodicity

Abstract

This paper is a follow-up to arXiv:2212.09639. We consider two algebraic settings of comodules over a coring and contramodules over a topological ring with a countable base of two-sided ideals. These correspond to two (noncommutative) algebraic geometry settings of certain kind of stacks and ind-affine ind-schemes. In the context of a coring $\mathcal{C}$ over a noncommutative ring $A$, we show that all $A$-flat $\mathcal{C}$-comodules are ${\aleph }_1$-directed colimits of $A$-countably presentable $A$-flat $\mathcal{C}$-comodules. In the context of a complete, separated topological ring $\mathfrak{R}$ with a countable base of neighborhoods of zero consisting of two-sided ideals, we prove that all flat $\mathfrak{R}$-contramodules are ${\aleph }_1$-directed colimits of countably presentable flat $\mathfrak{R}$-contramodules. We also describe arbitrary complexes, short exact sequences, and pure acyclic complexes of $A$-flat $\mathcal{C}$-comodules and flat $\mathfrak{R}$-contramodules as ${\aleph }_1$-directed colimits of similar complexes of countably presentable objects. The arguments are based on a very general category-theoretic technique going back to an unpublished 1977 preprint of Ulmer and rediscovered in arXiv:2310.16773. Applications to cotorsion periodicity and coderived categories of flat objects in the respective settings are discussed. In particular, in any acyclic complex of cotorsion $\mathfrak{R}$-contramodules, all the contramodules of cocycles are cotorsion.

arXiv commentLatex 2e with xy-pic; 44 pages, 1 commutative diagram; v.2: Sections 7-13 (about contramodules) added; v.3: Sections 3, 4, 10, and 11 completely rewritten based on arXiv:2310.16773, former Section 2 replaced by new Section 1, Remarks 3.2 and 10.3 added, the paper became much shorter with more results; v.4-v.5: small things corrected, references added and updated; v.6: several misprints corrected

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Cited by in corpus (5)

• Flat quasi-coherent sheaves as directed colimits, and quasi-coherent cotorsion periodicity
• Roos axiom holds for quasi-coherent sheaves
• Resolutions as directed colimits
• A contramodule generalization of Neeman's flat and projective module theorem
• The categories of corings and coalgebras over a ring are locally countably presentable