Types Of Serre Subcategories Of Grothendieck Categories And A Class Of Monomorphism Categories

Although recollements of abelian categories and exact sequences of abelian cat-egories can be similarly defined as in triangulated categories,there are still essential differences in triangulated categories and in abelian categories.For example,each functor in?left,right?recollements of triangulated categories is exact,while this is not the case in abelian categories.It is natural to ask what will happen if each functor involved in?left,right?recollements of abelian categories is also exact.On the other hand,Serre subcategories of abelian categories are the analogy of thick subcategories of triangulated categories.A Serre subcategory S of an abelian category A induces a quotient category A/S,and an exact sequence of abelian cate-gories ???.It is fundamental to ask if the embedding functor i and the quotient functor Q have a left adjoint functor or a right adjoint functor at the same time.If exist,is the new sequence also an exact sequence of abelian categories?Repeating this process,how long could this last.In views of this question,we introduce the notion of type of Serre subcategories.We realize that the above two questions are essentially same.When the abelian categories considered are Grothendieck categories,we get satisfied results.There-fore,the first part of this thesis focuses on the study of the Serre subcategories of Grothendieck categories,and the main results are as follows.In Chapter 3,we define bi-Giraud recollements of abelian categories and establish their relation with hereditary and cohereditary torsion pairs.Furthermore,we establish the relationships between the left?resp.right?recollements of abelian categories and strongly cohereditary?resp.hereditary?torsion pairs.In Chapter 4,two important observations are made.First,if the six functors in recollements of abelian categories are all exact,then the recollement are splitting so that the adjoint course can be continued forever.What's more,any left recolloments of Grothendieck categories always can be extended into a recollements.After these preparations,we get the classification of types of the Serre subcate-gories of Grothendieck categories.All the possible types of Serre subcategories of Grothendieck categories are one of the following seven forms?0,0?,?0,-1?,?1,-1?,?0,-2?,?1,-2?,?2,-1?,?+?,-??‘and for each type above,there exists a Serre subcategory of Grothendieck category of this type.The second part of this thesis is devoted to study a new kind of monomorphism categories.Given a selfinjective artin algebra A,we consider the category S_inj???,whose objects are the embeddings of finitely generated A-modules into injective A-modules.We show that S_inj???is a Frobenius category with Auslander-Reiten sequences,and that A-mod and S_inj???are stably equivalent;moreover,S_inj???has twice as many indecomposable injective objects as A-mod.