The Contravariant Finiteness Subcategories For The Right Orthogonal Category Of Tilting Modules

Let k be a algebraically closed field, A is finite dimensional algebraic over k,and modA is a category of all finitely generated right A modules. Migshita tell us that for the finite projective dimension and finitely generated tilting module T, B= EndT,Hom(T, -) and - (?) (DBT) give a correspondence be-tween the modules in T⊥ and the modules in ⊥(DBT);\let P<∞(modA) be the subcategory of modA consist of all modules of finite projective dimen-sion, Auslander and Retien had proved that if P<∞(modA) is contravariantly finite in modA,then the little dimension conjecture is true.Auslander and Retien has proved that[1] there is correspondence between the contravariantly finite resolving subcategory of modA and the tilting mod-ules.So if we consider the tilting module T and T⊥, similar as the definition of resolving subcategory and tilting modules in mod A, we define the T-resolving subcategory and T-tilting modules in T⊥,the main result of this passage is to prove that(Theorem 3.2.6) there is a correspondence between the contravari-antly finite T-resolving subcategories of T1 and the T-tilting modules contain inT⊥.In the last part,we will prove that(Theorem 4.2) if T<∞(T)={M ∈ T⊥|(?)0→Tn→Tn-1……T1→To→ M → 0,n ∈ N,Ti ∈ addT} is contravariantly finite in T⊥,then sup{pdM|M ∈ T<∞(T)} is finite.