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 The arguments are based on a very general category-theoretic technique going back to an unpublished 1977 preprint of Ulmer and rediscovered in arXiv:2310.16773.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.