On the telescope conjecture for module categories

In [H. Krause, O. Solberg, Applications of cotorsion pairs, J. London Math. Soc. 68 (2003) 631-6501, the Telescope Conjecture was formulated for the module category Mod R of an artin algebra R as follows: "If (f = (A, 8) is a complete hereditary cotorsion pair in Mod R with A and 8 closed under direct limits, then A = lim (A boolean AND mod R)". We extend this conjecture to arbitrary rings R, and show that it holds true if and only if the cotorsion pair C is of finite type. Then we prove the conjecture in the case when R is right noetherian and 8 has bounded injective dimension (thus, in particular, when C is any cotilting cotorsion pair). We also focus on the assumptions that A and B are closed under direct limits and on related closure properties, and detect several asymmetries in the properties of A and B. (c) 2007 Elsevier B.V. All rights reserved.