In this paper,we prove that if a triangulated category D admits a recollement relative to triangulated categories D' and D″,then the abelian category D/T admits a recollement relative to abelian categories D'...In this paper,we prove that if a triangulated category D admits a recollement relative to triangulated categories D' and D″,then the abelian category D/T admits a recollement relative to abelian categories D'/i(T) and D″/j(T) where T is a cluster tilting subcategory of D and satisfies i i (T) T,j j (T) T.展开更多
In this paper,we prove a reduction result on wide subcategories of abelian categories which is similar to the Calabi-Yau reduction,silting reduction andτ-tilting reduction.More precisely,if an abelian category A admi...In this paper,we prove a reduction result on wide subcategories of abelian categories which is similar to the Calabi-Yau reduction,silting reduction andτ-tilting reduction.More precisely,if an abelian category A admits a recollement relative to abelian categories A'and A'',which is denoted by(A',A,A'',i^(*),i_(*),i^(!),j_(!),j^(*),j_(*)),then the assignment C→j^(*)(C)defines a bijection between wide subcategories in A containing i_(*)(A')and wide subcategories in A''.Moreover,a wide subcategory C of A containing i_(*)(A')admits a new recollement relative to A'and j_(*)(C)which is induced from the original recollement.展开更多
Let T be a right exact functor from an abelian category B into another abelian category A.Then there exists a functor p from the product category A×B to the comma category(T↓A).In this paper,we study the propert...Let T be a right exact functor from an abelian category B into another abelian category A.Then there exists a functor p from the product category A×B to the comma category(T↓A).In this paper,we study the property of the extension closure of some classes of objects in(T↓A),the exactness of the functor p and the detailed description of orthogonal classes of a given class p(X,Y)in(T↓A).Moreover,we characterize when special precovering classes in abelian categories A and B can induce special precovering classes in(T↓A).As an application,we prove that under suitable conditions,the class of Gorenstein projective leftΛ-modules over a triangular matrix ringΛ=(R M 0 S)is special precovering if and only if both the classes of Gorenstein projective left R-modules and left S-modules are special precovering.Consequently,we produce a large variety of examples of rings such that the class of Gorenstein projective modules is special precovering over them.展开更多
Let A be an abelian category and M∈A.Then M is called a(D_(4))-object if,whenever A and B are subobjects of M with M=A⊕B and f:A→B is an epimorphism,Ker f is a direct summand of A.In this paper we give several equi...Let A be an abelian category and M∈A.Then M is called a(D_(4))-object if,whenever A and B are subobjects of M with M=A⊕B and f:A→B is an epimorphism,Ker f is a direct summand of A.In this paper we give several equivalent conditions of(D_(4))-objects in an abelian category.Among other results,we prove that any object M in an abelian category A is(D_(4))if and only if for every subobject K of M such that K is the intersection K_(1)∩K_(2)of perspective direct summands K_(1)and K_(2)of M with M=K_(1)+K_(2),every morphismφ:M→M/K can be lifted to an endomorphismθ:M→M in End _(A)(M).展开更多
In this paper we describe the category whose objects are principal ideals of a ring and morphisms are appropriate translations and it is shown that such a category is an abelian category. Further we discuss various pr...In this paper we describe the category whose objects are principal ideals of a ring and morphisms are appropriate translations and it is shown that such a category is an abelian category. Further we discuss various properties of such categories.展开更多
基金supported by National Natural Science Foundation of China (Grant No.10931006)the PhD Programs Foundation of Ministry of Education of China (Grant No.20060384002)the Scientific Research Foundation of Huaqiao University (Grant No.08BS506)
文摘In this paper,we prove that if a triangulated category D admits a recollement relative to triangulated categories D' and D″,then the abelian category D/T admits a recollement relative to abelian categories D'/i(T) and D″/j(T) where T is a cluster tilting subcategory of D and satisfies i i (T) T,j j (T) T.
基金This work was supported by the NSFC(Grant No.12201211)the China Scholarship Council(Grant No.202109710002).
文摘In this paper,we prove a reduction result on wide subcategories of abelian categories which is similar to the Calabi-Yau reduction,silting reduction andτ-tilting reduction.More precisely,if an abelian category A admits a recollement relative to abelian categories A'and A'',which is denoted by(A',A,A'',i^(*),i_(*),i^(!),j_(!),j^(*),j_(*)),then the assignment C→j^(*)(C)defines a bijection between wide subcategories in A containing i_(*)(A')and wide subcategories in A''.Moreover,a wide subcategory C of A containing i_(*)(A')admits a new recollement relative to A'and j_(*)(C)which is induced from the original recollement.
基金supported by National Natural Science Foundation of China (Grant Nos. 11671069 and 11771212)Zhejiang Provincial Natural Science Foundation of China (Grant No. LY18A010032)+1 种基金Qing Lan Project of Jiangsu Province and Jiangsu Government Scholarship for Overseas Studies (Grant No. JS2019-328)during a visit of the first author to Charles University in Prague with the support by Jiangsu Government Scholarship
文摘Let T be a right exact functor from an abelian category B into another abelian category A.Then there exists a functor p from the product category A×B to the comma category(T↓A).In this paper,we study the property of the extension closure of some classes of objects in(T↓A),the exactness of the functor p and the detailed description of orthogonal classes of a given class p(X,Y)in(T↓A).Moreover,we characterize when special precovering classes in abelian categories A and B can induce special precovering classes in(T↓A).As an application,we prove that under suitable conditions,the class of Gorenstein projective leftΛ-modules over a triangular matrix ringΛ=(R M 0 S)is special precovering if and only if both the classes of Gorenstein projective left R-modules and left S-modules are special precovering.Consequently,we produce a large variety of examples of rings such that the class of Gorenstein projective modules is special precovering over them.
基金Supported by Excellent Staff Room of Fuyang Teachers College(2013JCJS03)Natural Science Foundation of Fuyang Teachers College(2015FSKJ05)Natural Science Foundation of Universities in Anhui Province(2015KJ016)
文摘Let A be an abelian category and M∈A.Then M is called a(D_(4))-object if,whenever A and B are subobjects of M with M=A⊕B and f:A→B is an epimorphism,Ker f is a direct summand of A.In this paper we give several equivalent conditions of(D_(4))-objects in an abelian category.Among other results,we prove that any object M in an abelian category A is(D_(4))if and only if for every subobject K of M such that K is the intersection K_(1)∩K_(2)of perspective direct summands K_(1)and K_(2)of M with M=K_(1)+K_(2),every morphismφ:M→M/K can be lifted to an endomorphismθ:M→M in End _(A)(M).
文摘In this paper we describe the category whose objects are principal ideals of a ring and morphisms are appropriate translations and it is shown that such a category is an abelian category. Further we discuss various properties of such categories.
