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 ModR of an artin algebra RR as follows: “If C=(A,B)C=(A,B) is a complete hereditary cotorsion pair in ModR with AA and BB closed under direct limits, then A=lim⟶(A∩modR)”. We extend this conjecture to arbitrary rings RR, and show that it holds true if and only if the cotorsion pair CC is of finite type. Then we prove the conjecture in the case when RR is right noetherian and BB has bounded injective dimension (thus, in particular, when CC is any cotilting cotorsion pair). We also focus on the assumptions that AA and BB are closed under direct limits and on related closure properties, and detect several asymmetries in the properties of AA and BB.