Flat comodules and contramodules as directed colimits, and cotorsion periodicity

Abstract

This paper is a follow-up to Positselski and Št’ovíček (Flat quasi-coherent sheaves as directed colimits, and quasi-coherent cotorsion periodicity. Electronic preprint arXiv:2212.09639 [math.AG]). 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 The arguments are based on a very general category-theoretic technique going back to an unpublished 1977 preprint of Ulmer and rediscovered in Positselski (Notes on limits of accessible categories. Electronic preprint arXiv:2310.16773 [math.CT]). 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.