Tilting Module And Contravariantly Finitely Subcategories Closed Under Predecessors

Let A be an Artin algebra, modA be the category of finitely generated left A-modules, indA be the full subcategory of modA consists of the indecompos-able modules and C be the full subcategory of indA closed under predecessors.In this paper, we firstly prove that if addC contains an tilting module and it is Ext-injective in addC, then the additive full subcategory addC is contravariantly finite. Conversely, if addC is contravariantly finite, then the number of the indecomposable modules in C which is Ext-injective is equal to the number of the indecomposable projectives in C.Afterwards, we mainly discuss the union of two full categories C1and C2of indA which is closed under predecessors. The union of two full categories which is closed under predecessors is still closed under predecessors. For addi-tive full subcategories,we prove that the union of any two contravariantly finite subcategories is contravariantly.C1,C2are two subcategories of indA which is closed under predecessors,ε1,ε2are sets which is consists of all the indecom-posable Ext-injectives in C1,C2.we use ε1,ε2structure the sets ε which is consists of all the indecomposable Ext-injectives in C1U C2. Therefore,we get the module E corresponding to ε.If both addC1and addC2are contravariantly finite,add(C1U C2)=cogenE.Finally,paper [8] give an example that partial tilting have no complement in modd.A,but the proof is too brief, this paper gives a more detailed proof.