The countable Telescope Conjecture for module categories

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.