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 = under(lim, {long rightwards arrow}) (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 B.
On the telescope conjecture for module categories
ANGELERI, LIDIA;
2008-01-01
Abstract
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 = under(lim, {long rightwards arrow}) (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 B.
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