The Countable Telescope Conjecture for Module Categories

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. Motivated by the paper [30] of Krause and Solberg, the first author with Lidia Angeleri Hügel and Jan Trlifaj started in [4] an investigation of the Telescope Conjecture for Module Categories (TCMC) stated as follows (see Section 1 for unexplained terminology): Telescope Conjecture for Module Categories. Let R be a ring and (A,B) be a hereditary cotorsion pair in Mod-R with A and B closed under direct limits. Then A = lim −→ (A ∩mod-R). The term ‘Telescope Conjecture’ is used here because the particular case of TCMC when R is a self-injective artin algebra and (A,B) is a projective cotorsion pair was shown in [30] to be equivalent to the following telescope conjecture for compactly generated triangulated categories (in this case—for the stable module category over R) which originates in works of Bousfield [12] and Ravenel [38] and has been extensively studied by Krause in [29, 27]: Telescope Conjecture for Triangulated Categories. Every smashing localizing subcategory of a compactly generated triangulated category is generated by compact objects. Under some restrictions on homological dimensions of modules in the cotorsion pair (A,B), TCMC is known to hold. The first author and co-authors showed in [4] that the conclusion of TCMC amounts to saying that the given cotorsion pair is of Date: May 16, 2008. 2000 Mathematics Subject Classification. 16E30, 18E30 (primary), 03C60, 16D90, 18G25, 20K40 (secondary).