Generalized tilting functors and Ringel–Hall algebras

It is known from [M. Auslander, M.I. Platzeck, I. Reiten, Coxeter functors without diagrams, Trans. Amer. Math. Soc. 250 (1979) 1–46] and [C.M. Ringel, PBW-basis of quantum groups, J. Reine Angew. Math. 470 (1996) 51–85] that the Bernstein–Gelfand–Ponomarev reflection functors are special cases of tilting functors and these reflection functors induce isomorphisms between certain subalgebras of Ringel–Hall algebras. In [A. Wufu, Tilting functors and Ringel–Hall algebras, Comm. Algebra 33 (1) (2005) 343–348] the result from [C.M. Ringel, PBW-basis of quantum groups, J. Reine Angew. Math. 470 (1996) 51–85] is generalized to the tilting module case by giving an isomorphism between two Ringel–Hall subalgebras. In [J. Miyashita, Tilting Modules of Finite Projective Dimension, Math. Z. 193 (1986) 113–146] Miyashita generalized the tilting theory by introducing the tilting modules of finite projective dimension. In this paper the result in [A. Wufu, Tilting functors and Ringel–Hall algebras, Comm. Algebra 33 (1) (2005) 343–348] is generalized to the tilting modules of finite projective dimension.