On the telescope conjecture for module categories

In [H. Krause, O. Solberg, Applications of cotorsion pairs, J. London Math. Soc. 68 (2003) 631–650], the Telescope Conjecture was formulated for the module category Mod R of an artin algebra R as follows: “If C = (A,B) is a complete hereditary cotorsion pair in Mod R with A and B closed under direct limits, then A = lim −→ (A∩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 B 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