On The Properties Of Modules Related To Tilting Pairs

Tilting theory has play an important role in the representation theory of Artin algebra. Many methods have been brought forward through the pro-ces of further researches by domestic and internationnaol scholars. Tilting module and cotilting module, tilting pairs and generalized tilting pairs have been investigated. They also take a closely connection in equivalence theory, duality theory, approximation theory and homologically finite of subcategory of mod A.Chapter three concerns the properties of generalized tilting pairs. It is proved that Hom∧(C,T) is a generalized tilting right End∧(T)-module and as a right End∧(T)-module is selforthogonal which is induced by generalized tilting pair (C, T). The sufficient and necessary condition of the dual pair of generalized tilting pair (C, T) of mod A being still a generalized tilting pair is obained. It is showed that (1) if C, T∈modA, (C, T) is a generalized tilting pair in mod A if and only if (D(r),D(C)) is generalized tilting pair in mod ∧op; (2) if C, T∈modA, (C, T) is a generalized tilting pair in mod A if and only if (T*,C*) is a generalized tilting pair in mod Aop. It is also proved that if C, T ∈ mod∧,Γ= End∧(T), (C, T) is a generalized tilting pair, and at least one kernal of homomorphisms in the coresolution of module C in (?)T is∧TΓ-reflexive, then C is A∧TΓ-reflexive, Cω is such that Cω∈⊥T as a Γ-module.Chapter four pays attention to the approximation properties of cotilting pairs. It is proved that there exists a left T1-approximation of N in modA and N has a right addT-approximation. It is shown that if (C, T) is a cotilting pair, C is a cogenerater of modA and ⊥N-dim∧(C)<∞ for all N ∈mod∧, then T⊥-is covariantly finite subcategory of modA, addT is contravariantly finite subcategory of modA.