The countable Telescope Conjecture for module categories

TL;DR

This paper investigates a countable version of the Telescope Conjecture for module categories, providing new characterizations of hereditary cotorsion pairs and their modules under certain closure conditions.

Contribution

It proves a modified conjecture replacing 'finite' with 'countable', characterizes modules in cotorsion pairs, and explores analogies with triangulated categories.

Findings

Hereditary cotorsion pairs are generated by strongly countably presented modules.

Modules in B are characterized via morphisms between finitely presented modules.

Cotorsion pairs are cogenerated by a single pure-injective module under certain conditions.

Abstract

By the Telescope Conjecture for Module Categories, we mean the following claim: "Let R be any ring and (A, B) be a hereditary cotorsion pair in Mod-R with A and B closed under direct limits. Then (A, B) is of finite type." We prove a modification of this conjecture with the word 'finite' replaced by 'countable'. We show that a hereditary cotorsion pair (A, B) of modules over an arbitrary ring R is generated by a set of strongly countably presented modules provided that B is closed under unions of well-ordered chains. We also characterize the modules in B and the countably presented modules in A in terms of morphisms between finitely presented modules, and show that (A, B) is cogenerated by a single pure-injective module provided that A is closed under direct limits. Then we move our attention to strong analogies between cotorsion pairs in module categories and localizing pairs in compactly generated triangulated categories.