(Co)torpion Pairs In (n+2)-angulated Categories And N-exangulated Categories

This thesis mainly studies(co)torsion pairs in(n+2)-angulated categories and n-exangulated categories and the relationship between(co)torsion pairs in these categories and the(n+2)-angulated quotient categories induced by them.In the first part,we define the notion of the(co)torsion pairs in an(n+2)-angulated category C(see Definition 31)and Ext-projective,Ext-injective objects in an subcategory X(see Definition 3.6)to study several special cotorsion pairs.Under some conditions on subcategories Z and D in C,we construct an(n+2)-angulated quotient category U:=Z/D with the shift functor T following[8].And in this case,we get a general result(see Theorem 5.9):If(Z,Z)is a D-mutation pair,Z is closed under n-extensions,and C(D,∑D)=0,then given two sub-categories X,y of C,(1)if(X,∑y)is a torsion pair in C with D(?)C X,y(?)Z,then(X,Ty)is a torsion pair in U;(2)if ∑1-ny)is a cotorsion pair in C with D(?)X,y(?)Z,then(X,T1-ny)is a cotorsion pair in U.In the second part,we replace the(n+2)-angulated category with an n-exangulated category,still denoted by C.We define the notion of the(co)torsion pairs in C(see Definition 4.1)and E-projective,E-injective objects in an subcat-egory X(see Definition 4.5)to study several special cotorsion pairs.Following[10],we construct an(n+2)-angulated quotient category U:=Z/D with the shift functor(1).In this case,we get a general result(see Theorem 5.18):If(Z,Z)is a D-mutation pair and E(D,D)=0,then given two subcategories X,y of C,(1)if(X,y)is a torsion pair in C with D(?)X,Y(?)Z,then(X,Y<1>)is a torsion pair in U;(2)if(X,y)is a cotorsion pair in C with D(?)X,Y(?)Z,then(X,Y<1-n))is a cotorsion pair in U.