We show that a tilting module T over a ring R admits an exact sequence 0→ R→T0→T1→0 such that T0, T1 ∈ Add(T) and HomR(T1, T0) = 0 if and only if T has the form S ⊕ S/R for some injective ring epimorphism λ R→ S with the property that Tor1R (S, S) = 0 and pdSR ≤ 1. We then study the case where λ is a universal localization in the sense of Schofield (1985). Using results by Crawley- Boevey (Proc Lond Math Soc (3) 62(3):490-508, 1991), we give applications to tame hereditary algebras and hereditary noetherian prime rings. In particular, we show that every tilting module over a Dedekind domain or over a classical maximal order arises from universal localization. © Springer Science + Business Media B.V. 2009.
- Dedekind domain
- Hereditary noetherian prime ring
- Ring epimorphism
- Tilting module
- Universal localization