The countable Telescope Conjecture for module categories

• Autores: Jan Saroch, Jan Stovícek
• Localización: Advances in mathematics, ISSN 0001-8708, Vol. 219, Nº 3, 2008, págs. 1002-1036
• Idioma: inglés
• DOI: 10.1016/j.aim.2008.05.012
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• Resumen

  • 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.