Let T=(A0 UB)be a triangular matrix ring with A,B rings and U a B-A-bimodule.We construct resolving subcategories of T-Mod from those of A-Mod and B-Mod.Then we give an estimate of the global resolving resolution dime...Let T=(A0 UB)be a triangular matrix ring with A,B rings and U a B-A-bimodule.We construct resolving subcategories of T-Mod from those of A-Mod and B-Mod.Then we give an estimate of the global resolving resolution dimension of T in terms of that of A and of B.Some applications of these results are given.展开更多
In this paper,we generalize the idea of Song,Zhao and Huang[Czechoslov.Math.J.,70,483±504(2020)]and introduce the notion of right(left)Gorenstein subcategory rg(l,∂)(lg(l,D)),relative to two additive full subcate...In this paper,we generalize the idea of Song,Zhao and Huang[Czechoslov.Math.J.,70,483±504(2020)]and introduce the notion of right(left)Gorenstein subcategory rg(l,∂)(lg(l,D)),relative to two additive full subcategoriesφand∂of an abelian category A.Under the assumption thatφ⊆∂,we prove that the right Gorenstein subcategory rg(l,D)possesses many nice properties that it is closed under extensions,kernels of epimorphisms and direct summands.Whenφ⊆Dandφ⊥D,we show that the right Gorenstein subcategory rg(l,D)admits some kind of stability.Then we discuss a resolution dimension for an object in A,called rg(l,D)-projective dimension.Finally,we prove that if(U,V)is a hereditary cotorsion pair with kernelφhas enough injectives,such that U⊆Dand U⊥∂,then(rg(l,D),φφ)is a weak Auslander±Buchweitz context,whereφis the subcategory of A consisting of objects with finiteφ-projective dimension.展开更多
We show that the torsion module Tor_(j)^(R)(R/a,H_(a)^(i)(X))is in a Serre subcategory for the bounded below R-complex X.In addition,we prove the isomorphism Tor_(s-t)^(R)(R/a,X)≅Tor_(s)^(R)(R/a,H_(a)^(t)(X))in some c...We show that the torsion module Tor_(j)^(R)(R/a,H_(a)^(i)(X))is in a Serre subcategory for the bounded below R-complex X.In addition,we prove the isomorphism Tor_(s-t)^(R)(R/a,X)≅Tor_(s)^(R)(R/a,H_(a)^(t)(X))in some case.As an application,the Betti number of a complex X in a prime ideal p can be computed by the Betti number of the local cohomology modules of X in p.展开更多
U_(S)-admitting spaces,which were introduced by Heckmann,enjoy many nice properties similar to those of the extensively studied well-filtered spaces.In this paper,we present a direct construction of the U_(S)-admittin...U_(S)-admitting spaces,which were introduced by Heckmann,enjoy many nice properties similar to those of the extensively studied well-filtered spaces.In this paper,we present a direct construction of the U_(S)-admitting reflections by using U_(S)-admitting determined sets.展开更多
As a generalization of tilting pair, which was introduced by Miyashita, the notion of silting pair is introduced in this paper. The authors extend a characterization of tilting modules given by Bazzoni to silting pair...As a generalization of tilting pair, which was introduced by Miyashita, the notion of silting pair is introduced in this paper. The authors extend a characterization of tilting modules given by Bazzoni to silting pairs, and prove that there is a one-to-one correspondence between equivalent classes of silting pairs and certain subcategories which satisfy some conditions.Furthermore, the authors also give a bijection between equivalent class of silting pairs and bounded above co-t-structure.展开更多
Let A be an abelian category,and(X,Z,Y)be a complete hereditary cotorsion triple.We introduce the definition of n-Y-cotilting subcategories of A,and give a characterization of n-Y-cotilting subcategories,which is simi...Let A be an abelian category,and(X,Z,Y)be a complete hereditary cotorsion triple.We introduce the definition of n-Y-cotilting subcategories of A,and give a characterization of n-Y-cotilting subcategories,which is similar to Bazzoni characterization of n-cotilting modules.As an application,we prove that if GP is n-GI-cotilting over a virtually Gorenstein ring R,then R is an n-Gorenstein ring,where GP denotes the subcategory of Gorenstein projective R-modules and GI denotes the subcategory of Gorenstein injective R-modules.Furthermore,we investigate n-costar subcategories over arbitrary ring R,and the relationship between n-Icotilting subcategories with respect to cotorsion triple(P,R-Mod,I)and n-costar subcategories,where P denotes the subcategory of projective left R-modules and I denotes the subcategory of injective left R-modules.展开更多
In this paper, we introduce the definition of n-star subcategories, which is a generalization of n-star modules and n-C-star modules. We give some characterizations of n-star subcategories, and prove that M is an n-P-...In this paper, we introduce the definition of n-star subcategories, which is a generalization of n-star modules and n-C-star modules. We give some characterizations of n-star subcategories, and prove that M is an n-P-tilting subcategory with respect to cotorsion triple(P, R-Mod, I), if and only if M is an n-star subcategory with I ■Pres^(n)(M), where P denotes the subcategory of projective left R-modules and I denotes the subcategory of injective left R-modules.展开更多
Let A be an abelian category,T a self-orthogonal subcategory of A and each object in T admit finite projective and injective dimensions.If the left Gorenstein subcategory lG(T)equals to the right orthogonal class of T...Let A be an abelian category,T a self-orthogonal subcategory of A and each object in T admit finite projective and injective dimensions.If the left Gorenstein subcategory lG(T)equals to the right orthogonal class of T and the right Gorenstein subcategory rG(T)equals to the left orthogonal class of T,we prove that the Gorenstein subcategory G(T)equals to the intersection of the left orthogonal class of T and the right orthogonal class of T,and prove that their stable categories are triangle equivalent to the relative singularity category of A with respect to T.As applications,let R be a left Noetherian ring with finite left self-injective dimension and _(R)C_(S) a semidualizing bimodule,and let the supremum of the flat dimensions of all injective left R-modules be finite.We prove that if RC has finite injective(or flat)dimension and the right orthogonal class of C contains R,then there exists a triangle-equivalence between the intersection of C-Gorenstein projective modules and Bass class with respect to C,and the relative singularity category with respect to C-projective modules.Some classical results are generalized.展开更多
As a non-trivial generalization of quasi-resolving subcategories,the notion of Ext-quasi-resolving subcategories of an abelian category is introduced.Moreover,we give a general example,which is not a quasi-resolving s...As a non-trivial generalization of quasi-resolving subcategories,the notion of Ext-quasi-resolving subcategories of an abelian category is introduced.Moreover,we give a general example,which is not a quasi-resolving subcategory,and the homological theory of Ext-quasi-resolving subcategories is studied.In particular,we generalize many results on the resolving subcategories.展开更多
Let C be a triangulated category. We define m-term subcategories on C induced by n-rigid subcategories, which are extriangulated subcategories of C. Then we give a one-to-one correspondence between cotorsion pairs on ...Let C be a triangulated category. We define m-term subcategories on C induced by n-rigid subcategories, which are extriangulated subcategories of C. Then we give a one-to-one correspondence between cotorsion pairs on 2-term subcategories G and support τ-tilting subcategories on an abelian quotient of G. If an m-term subcategory is induced by a co-t-structure, then we have a one-to-one correspondence between cotorsion pairs on it and cotorsion pairs on C under certain conditions.展开更多
In this paper it is proved that for all completely distributive lattices L. the category of L-fuzzifying topological spaces can be wmbedded in the category of L-topological spaces (stratified Chang-Goguen spaces) as a...In this paper it is proved that for all completely distributive lattices L. the category of L-fuzzifying topological spaces can be wmbedded in the category of L-topological spaces (stratified Chang-Goguen spaces) as a simultaneously bireflective and bicoreflective full subcategory.展开更多
Let L be a meet continuous lattice. It is proved that the category Top of topological spaces can be embedded in the category of strati?ed L-topological spaces as a concretely both re?ective and core?ective full subcat...Let L be a meet continuous lattice. It is proved that the category Top of topological spaces can be embedded in the category of strati?ed L-topological spaces as a concretely both re?ective and core?ective full subcategory if and only if L is a continuous lattice.展开更多
Let A be a finite-dimensional algebra over an algebraically closed field k,ε the category of all exact sequences in A-rood, Mp （respectively, Ml） the full subcategory of C consisting of those objects with projecti...Let A be a finite-dimensional algebra over an algebraically closed field k,ε the category of all exact sequences in A-rood, Mp （respectively, Ml） the full subcategory of C consisting of those objects with projective （respectively, injective） middle terms. It is proved that Mp （respectively, MI） is contravariantly finite （respectively, covariantly finite） in ε. As an application, it is shown that Mp = MI is functorially finite and has Auslander-Reiten sequences provided A is selfinjective. Keywords category of exact sequences, contravariantly finite subcategory, functorially finite subcategory Auslander-Reiten sequences, selfinjective algebra展开更多
Let C be a triangulated category which has Auslander-Reiten triangles, and Ra functorially finite rigid subcategory of C. It is well known that there exist Auslander-Reiten sequences in rood R. In this paper, we give ...Let C be a triangulated category which has Auslander-Reiten triangles, and Ra functorially finite rigid subcategory of C. It is well known that there exist Auslander-Reiten sequences in rood R. In this paper, we give explicitly the relations between the Auslander-Reiten translations, sequences in mod R and the Auslander-Reiten functors, triangles in C, respectively. Furthermore, if T is a cluster-tilting subcategory of C and mod T- is a Frobenius category, we also get the Auslander-Reiten functor and the translation functor of mod T- corresponding to the ones in C. As a consequence, we get that if the quotient of a d-Calabi-Yau triangulated category modulo a cluster tilting subcategory is Probenius, then its stable category is （2d-1）-Calabi-Yau. This result was first proved by Keller and Reiten in the case d= 2, and then by Dugas in the general case, using different methods. 2010 Mathematics Subject Classification： 16G20, 16G70展开更多
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
Extriangulated category was introduced by H.Nakaoka and Y.Palu to give a unification of properties in exact categories anjd triangulated categories.A notion of tilting(resp.,cotilting)subcategories in an extriangulate...Extriangulated category was introduced by H.Nakaoka and Y.Palu to give a unification of properties in exact categories anjd triangulated categories.A notion of tilting(resp.,cotilting)subcategories in an extriangulated category is defined in this paper.We give a Bazzoni characterization of tilting(resp.,cotilting)subcategories and obtain an Auslander-Reiten correspondence between tilting(resp.,cotilting)subcategories and coresolving covariantly(resp.,resolving contravariantly)finite subcatgories which are closed under direct summands and satisfy some cogenerating(resp.,generating)conditions.Applications of the results are given:we show that tilting(resp.,cotilting)subcategories defined here unify many previous works about tilting modules(subcategories)in module categories of Artin algebras and in abelian categories admitting a cotorsion triples;we also show that the results work for the triangulated categories with a proper class of triangles introduced by A.Beligiannis.展开更多
In Enochs'relative homological dimension theory occur the(co)resolvent and(co)proper dimensions,which are defined by proper and coproper resolutions constructed by precovers and preenvelopes,respectively.Recently,...In Enochs'relative homological dimension theory occur the(co)resolvent and(co)proper dimensions,which are defined by proper and coproper resolutions constructed by precovers and preenvelopes,respectively.Recently,some authors have been interested in relative homological dimensions defined by just exact sequences.In this paper,we contribute to the investigation of these relative homological dimensions.First we study the relation between these two kinds of relative homological dimensions and establish some transfer results under adjoint pairs.Then relative global dimensions are studied,which lead to nice characterizations of some properties of particular cases of self-orthogonal subcategories.At the end of this paper,relative derived functors are studied and generalizations of some known results of balance for relative homology are established.展开更多
The notion of a tilting pair Miyashita in 2001. It is a useful tool in cotorsion pairs related to a fixed tilting （covariantly） finite subcategory and a tilting pair were given in this paper. over artin algebras was...The notion of a tilting pair Miyashita in 2001. It is a useful tool in cotorsion pairs related to a fixed tilting （covariantly） finite subcategory and a tilting pair were given in this paper. over artin algebras was introduced by the tilting theory. Approximations and pair were discussed. A eontravariantly eotorsion pair associated with a fixed展开更多
For any n 3, let R(n) denote the root category of finite-dimensional nilpotent representations of cyclic quiver with n vertices. In the present paper, we prove that R(n-1) is triangle-equivalent to the subcategory of ...For any n 3, let R(n) denote the root category of finite-dimensional nilpotent representations of cyclic quiver with n vertices. In the present paper, we prove that R(n-1) is triangle-equivalent to the subcategory of fixed points of certain left (or right) mutation in R(n). As an application, it is shown that the affine Kac-Moody algebra of type n-2 is isomorphic to a Lie subalgebra of the Kac-Moody algebra of type n-1.展开更多
文摘Let T=(A0 UB)be a triangular matrix ring with A,B rings and U a B-A-bimodule.We construct resolving subcategories of T-Mod from those of A-Mod and B-Mod.Then we give an estimate of the global resolving resolution dimension of T in terms of that of A and of B.Some applications of these results are given.
基金Supported by National Natural Science Foundation of China(Grant No.11971225)。
文摘In this paper,we generalize the idea of Song,Zhao and Huang[Czechoslov.Math.J.,70,483±504(2020)]and introduce the notion of right(left)Gorenstein subcategory rg(l,∂)(lg(l,D)),relative to two additive full subcategoriesφand∂of an abelian category A.Under the assumption thatφ⊆∂,we prove that the right Gorenstein subcategory rg(l,D)possesses many nice properties that it is closed under extensions,kernels of epimorphisms and direct summands.Whenφ⊆Dandφ⊥D,we show that the right Gorenstein subcategory rg(l,D)admits some kind of stability.Then we discuss a resolution dimension for an object in A,called rg(l,D)-projective dimension.Finally,we prove that if(U,V)is a hereditary cotorsion pair with kernelφhas enough injectives,such that U⊆Dand U⊥∂,then(rg(l,D),φφ)is a weak Auslander±Buchweitz context,whereφis the subcategory of A consisting of objects with finiteφ-projective dimension.
基金Natural Science Foundation of Gansu Province(23JRRA866)Higher Education Innovation Fund of Gansu Provincial Department of Education(2025A-132)+1 种基金University-level Scientific Research and Innovation Project of Gansu University of Political Science and Law(GZF2024XQN16)Youth Foundation of Lanzhou Jiaotong University(2023023)。
文摘We show that the torsion module Tor_(j)^(R)(R/a,H_(a)^(i)(X))is in a Serre subcategory for the bounded below R-complex X.In addition,we prove the isomorphism Tor_(s-t)^(R)(R/a,X)≅Tor_(s)^(R)(R/a,H_(a)^(t)(X))in some case.As an application,the Betti number of a complex X in a prime ideal p can be computed by the Betti number of the local cohomology modules of X in p.
基金Supported by the National Natural Science Foundation of China(Grant No.12571507)。
文摘U_(S)-admitting spaces,which were introduced by Heckmann,enjoy many nice properties similar to those of the extensively studied well-filtered spaces.In this paper,we present a direct construction of the U_(S)-admitting reflections by using U_(S)-admitting determined sets.
基金Supported by the National Natural Science Foundation of China (Grant No. 11801004)the Top Talent Project of AHPU in 2020 (Grants No. S022021055)。
文摘As a generalization of tilting pair, which was introduced by Miyashita, the notion of silting pair is introduced in this paper. The authors extend a characterization of tilting modules given by Bazzoni to silting pairs, and prove that there is a one-to-one correspondence between equivalent classes of silting pairs and certain subcategories which satisfy some conditions.Furthermore, the authors also give a bijection between equivalent class of silting pairs and bounded above co-t-structure.
基金Supported by Research Project in Institutions of Higher Learning in Gansu Province(Grant No.2019B-224)Innovation Fund Project of Colleges and Universities in Gansu Province(Grant No.2020A-277)。
文摘Let A be an abelian category,and(X,Z,Y)be a complete hereditary cotorsion triple.We introduce the definition of n-Y-cotilting subcategories of A,and give a characterization of n-Y-cotilting subcategories,which is similar to Bazzoni characterization of n-cotilting modules.As an application,we prove that if GP is n-GI-cotilting over a virtually Gorenstein ring R,then R is an n-Gorenstein ring,where GP denotes the subcategory of Gorenstein projective R-modules and GI denotes the subcategory of Gorenstein injective R-modules.Furthermore,we investigate n-costar subcategories over arbitrary ring R,and the relationship between n-Icotilting subcategories with respect to cotorsion triple(P,R-Mod,I)and n-costar subcategories,where P denotes the subcategory of projective left R-modules and I denotes the subcategory of injective left R-modules.
基金Supported by the 2020 Scientific Research Projects in Universities of Gansu Province (Grant No. 2020A-277)。
文摘In this paper, we introduce the definition of n-star subcategories, which is a generalization of n-star modules and n-C-star modules. We give some characterizations of n-star subcategories, and prove that M is an n-P-tilting subcategory with respect to cotorsion triple(P, R-Mod, I), if and only if M is an n-star subcategory with I ■Pres^(n)(M), where P denotes the subcategory of projective left R-modules and I denotes the subcategory of injective left R-modules.
基金Supported by the Project of Natural Science Foundation of Changzhou College of Information Technology(Grant No.CXZK202204Y)the Project of Youth Innovation Team of Universities of Shandong Province(Grant No.2022KJ314)。
文摘Let A be an abelian category,T a self-orthogonal subcategory of A and each object in T admit finite projective and injective dimensions.If the left Gorenstein subcategory lG(T)equals to the right orthogonal class of T and the right Gorenstein subcategory rG(T)equals to the left orthogonal class of T,we prove that the Gorenstein subcategory G(T)equals to the intersection of the left orthogonal class of T and the right orthogonal class of T,and prove that their stable categories are triangle equivalent to the relative singularity category of A with respect to T.As applications,let R be a left Noetherian ring with finite left self-injective dimension and _(R)C_(S) a semidualizing bimodule,and let the supremum of the flat dimensions of all injective left R-modules be finite.We prove that if RC has finite injective(or flat)dimension and the right orthogonal class of C contains R,then there exists a triangle-equivalence between the intersection of C-Gorenstein projective modules and Bass class with respect to C,and the relative singularity category with respect to C-projective modules.Some classical results are generalized.
基金Supported by the Natural Science Foundation of Universities of Anhui(2023AH050950,2023AH050904)the Top Talent Project of AHPU in 2020(S022021055)+2 种基金the National Natural Science Foundation of China(11801004,12101003,12301042)the Natural Science Foundation of Anhui Province(2108085QA07)the Startup Foundation for Introducing Talent of AHPU(2020YQQ067,2022YQQ097).
文摘As a non-trivial generalization of quasi-resolving subcategories,the notion of Ext-quasi-resolving subcategories of an abelian category is introduced.Moreover,we give a general example,which is not a quasi-resolving subcategory,and the homological theory of Ext-quasi-resolving subcategories is studied.In particular,we generalize many results on the resolving subcategories.
基金supported by the National Natural Science Foundation of China(Grant No.12171397)Panyue Zhou is supported by the National Natural Science Foundation of China(Grant No.12371034)by the Hunan Provincial Natural Science Foundation of China(Grant No.2023JJ30008)。
文摘Let C be a triangulated category. We define m-term subcategories on C induced by n-rigid subcategories, which are extriangulated subcategories of C. Then we give a one-to-one correspondence between cotorsion pairs on 2-term subcategories G and support τ-tilting subcategories on an abelian quotient of G. If an m-term subcategory is induced by a co-t-structure, then we have a one-to-one correspondence between cotorsion pairs on it and cotorsion pairs on C under certain conditions.
基金This work is supported by the Natural Science Foundation of Chinathe Foundation for Fellows Returned from Abroadthe Mathematical Center of the Education Ministry of China
文摘In this paper it is proved that for all completely distributive lattices L. the category of L-fuzzifying topological spaces can be wmbedded in the category of L-topological spaces (stratified Chang-Goguen spaces) as a simultaneously bireflective and bicoreflective full subcategory.
基金Project supported by the 973 Project of the Ministry of Science and Technology of China(No.2002cb312200)the Excellent Young Teachers Program of the Ministry of Education of Chinaand Huo Yingdong Education Foundation.
文摘Let L be a meet continuous lattice. It is proved that the category Top of topological spaces can be embedded in the category of strati?ed L-topological spaces as a concretely both re?ective and core?ective full subcategory if and only if L is a continuous lattice.
基金supported by National Natural Science Foundation of China(Grant No.11271257)National Science Foundation of Shanghai Municiple(Granted No.13ZR1422500)
文摘Let A be a finite-dimensional algebra over an algebraically closed field k,ε the category of all exact sequences in A-rood, Mp （respectively, Ml） the full subcategory of C consisting of those objects with projective （respectively, injective） middle terms. It is proved that Mp （respectively, MI） is contravariantly finite （respectively, covariantly finite） in ε. As an application, it is shown that Mp = MI is functorially finite and has Auslander-Reiten sequences provided A is selfinjective. Keywords category of exact sequences, contravariantly finite subcategory, functorially finite subcategory Auslander-Reiten sequences, selfinjective algebra
文摘Let C be a triangulated category which has Auslander-Reiten triangles, and Ra functorially finite rigid subcategory of C. It is well known that there exist Auslander-Reiten sequences in rood R. In this paper, we give explicitly the relations between the Auslander-Reiten translations, sequences in mod R and the Auslander-Reiten functors, triangles in C, respectively. Furthermore, if T is a cluster-tilting subcategory of C and mod T- is a Frobenius category, we also get the Auslander-Reiten functor and the translation functor of mod T- corresponding to the ones in C. As a consequence, we get that if the quotient of a d-Calabi-Yau triangulated category modulo a cluster tilting subcategory is Probenius, then its stable category is （2d-1）-Calabi-Yau. This result was first proved by Keller and Reiten in the case d= 2, and then by Dugas in the general case, using different methods. 2010 Mathematics Subject Classification： 16G20, 16G70
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the National Natural Science Foundation of China(Grant No.11671221).
文摘Extriangulated category was introduced by H.Nakaoka and Y.Palu to give a unification of properties in exact categories anjd triangulated categories.A notion of tilting(resp.,cotilting)subcategories in an extriangulated category is defined in this paper.We give a Bazzoni characterization of tilting(resp.,cotilting)subcategories and obtain an Auslander-Reiten correspondence between tilting(resp.,cotilting)subcategories and coresolving covariantly(resp.,resolving contravariantly)finite subcatgories which are closed under direct summands and satisfy some cogenerating(resp.,generating)conditions.Applications of the results are given:we show that tilting(resp.,cotilting)subcategories defined here unify many previous works about tilting modules(subcategories)in module categories of Artin algebras and in abelian categories admitting a cotorsion triples;we also show that the results work for the triangulated categories with a proper class of triangles introduced by A.Beligiannis.
基金The second and fourth authors were partially supported by the grant MTM2014-54439-P from Ministerio de Economia y CompetitividadThe third author was partially supported by NSFC(11771202).
文摘In Enochs'relative homological dimension theory occur the(co)resolvent and(co)proper dimensions,which are defined by proper and coproper resolutions constructed by precovers and preenvelopes,respectively.Recently,some authors have been interested in relative homological dimensions defined by just exact sequences.In this paper,we contribute to the investigation of these relative homological dimensions.First we study the relation between these two kinds of relative homological dimensions and establish some transfer results under adjoint pairs.Then relative global dimensions are studied,which lead to nice characterizations of some properties of particular cases of self-orthogonal subcategories.At the end of this paper,relative derived functors are studied and generalizations of some known results of balance for relative homology are established.
文摘The notion of a tilting pair Miyashita in 2001. It is a useful tool in cotorsion pairs related to a fixed tilting （covariantly） finite subcategory and a tilting pair were given in this paper. over artin algebras was introduced by the tilting theory. Approximations and pair were discussed. A eontravariantly eotorsion pair associated with a fixed
基金supported by National Natural Science Foundation of China (Grant Nos.10931006, 10926041)
文摘For any n 3, let R(n) denote the root category of finite-dimensional nilpotent representations of cyclic quiver with n vertices. In the present paper, we prove that R(n-1) is triangle-equivalent to the subcategory of fixed points of certain left (or right) mutation in R(n). As an application, it is shown that the affine Kac-Moody algebra of type n-2 is isomorphic to a Lie subalgebra of the Kac-Moody algebra of type n-1.
