The countable Telescope Conjecture for module categories

Details The countable Telescope Conjecture for module categories The countable Telescope Conjecture for module categories

2008-01-25

arxiv.org/abs/0801.3936v2

Adv. Math. 219 (2008) 1002-1036

Mathematics

Rings and Algebras

31 pages; minor changes, typos corrected, references added

Scientific paper

10.1016/j.aim.2008.05.012

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.

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