Structural Properties Of A Kind Of Generalized Tilting Modules

Wakamatsu tilting module is a generalization of tilting module. It is a natural thing to investigate the relationship between Wakamatsu tilting mod-ules and tilting modules. The Wakamatsu tilting conjecture implies that a Wakamatsu tilting module of finite projective dimension is a tilting module. Mantese and Reiten showed some conclusions about the conjecture in [3], and they also discussed the topic of its close relationship to well known homological conjectures, like the generalized Nakayama conjecture.During the study to it, a naturally idea is that whether all the properties of tilting module can fit Wakamatsu tilting module with finite projective di-mension. Here we will try to summarize the properties and try to prove them systematically, for the aim of learning.This study proved the conclusion that Wakamatsu tilting modules of finite projective dimension have the same properties as tilting modules: For an arbitrary ring R, if T ∈∈ ModR is a Wakamatsu tilting module with finite projective dimension, then:(a) (⊥X)⊥= X, that’s [⊥(T⊥)]⊥= T⊥.(b) For any X ∈ X, there exists an exact sequence 0→K→T’→ X→0 where T’ ∈ AddT and K ∈ X(c) AddT= X∩⊥X(d) For any X ∈ X of finite projective dimension, there exists a long exact sequence 0→Tn+1→Tn→Tn-1→……T0…X→0where for i=0,….,n+1 we have Ti∈AddT.