The Right Gorenstein Subcategory rg(l,D)

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.

关于余挠三元组的n-星子范畴和n-P-倾斜子范畴

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.

∞-Tilting Subcategories in Extriangulated Categories

In this paper, the authors introduce a new definition of ∞-tilting(resp. cotilting) subcategories with infinite projective dimensions(resp. injective dimensions) in an extriangulated category. They give a Bazzoni characterization of ∞-tilting(resp. cotilting)subcategories. Also, they obtain a partial Auslander-Reiten correspondence between ∞-tilting(resp. cotilting) subcategories and coresolving(resp. resolving) subcategories with an E-projective generator(resp. E-injective cogenerator) in an extriangulated category.

The Extension Dimension of Subcategories and Recollements of Abelian Categories

We investigate the behavior of the extension dimension of subcategories of abelian categories under recollements.LetΛ',Λ,Λ"be art in algebras such that(modΛ',mod A,modΛ")is a recollement,and let D'and D"be subcategories of modΛand modΛ"respectively.For any n,m≥0,under some conditions,we get dimΩ^(k)(D)≤dimΩ^(n)(D')+dimΩ^(m)(D")+1,where k=max{m,n}and D is the subcategory of modΛglued by D'and D";moreover,we give a sufficient condition such that the converse inequality holds true.As applications,some results for Igusa-Todorov subcategories and syzygy finite sub categories are obtained.

Homological Transfer between Additive Categories and Higher Differential Additive Categories

Given an additive category C and an integer n≥2.The higher differential additive category consists of objects X in C equipped with an endomorphism ϵ_(X)satisfying ϵ_(X)^(n).Let R be a,finite-dimensional basic algebra over an algebraically closed field and T the augmenting functor from the category of finitely generated left R-modules to that of finitely generated left R/(t^(n))-modules.It is proved that a finitely generated left R-module M isτ-rigid(respectively,(support)τ-tilting,almost completeτ-tilting)if and only if so is T(M)as a left R[t]/(t^(n))-module.Moreover,R isτm-selfinjective if and only if so is R[t]/(t^(n)).

Reduction of Wide Subcategories and Recollements

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.

Indices and c-vectors in extriangulated categories

Let C be an extriangulated category and T be any n-cluster tilting subcategory of C.We consider the index with respect to T and introduce the index Grothendieck group of T.Using the index,we prove that the index Grothendieck group of T is isomorphic to the Grothendieck group of C,which implies that the index Grothendieck groups of any two n-cluster tilting subcategories are isomorphic.In particular,we show that the split Grothendieck groups of any two 2-cluster tilting subcategories are isomorphic.Then we develop a general framework for c-vectors of 2-Calabi-Yau extriangulated categories with respect to arbitrary 2-cluster tilting subcategories.We show that the c-vectors have the sign-coherence property and provide some formulas for calculating c-vectors.

n-exact Categories Arising from n-exangulated Categories

Let l be a Krull-Schmidt n-exangulated category and A be an n-extension closed subcategory of l.Then A inherits the n-exangulated structure from the given n-exangulated category in a natural way.This construction gives n-exangulated categories which are neither n-exact categories in the sense of Jasso nor(n+2)-angulated categories in the sense of Geiss-Keller-Oppermann in general.Furthermore,we also give a sufficient condition on when an n-exangulated category A is an n-exact category.These results generalize work by Klapproth and Zhou.

On the Reflectivity and Coreflectivity of L-Fuzzifying Topological Spaces in L-Topological Spaces

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.

ON THE EMBEDDING OF TOP IN THE CATEGORY OF STRATIFIED L-TOPOLOGICAL SPACES

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.

Auslander-Reiten Sequences or Triangles Related to Rigid Subcategories

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

From recollement of triangulated categories to recollement of abelian categories

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.

Tilting subcategories in extriangulated categories

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.

Subcategories of fixed points of mutations in root categories with type _(n-1)

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.

On Proper and Exact Relative Homological Dimensions

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.

Subcategories of Fixed Points of Mutations by Exceptional Objects in Triangulated Categories

We first prove that the subcategory of fixed points of mutation determined by an excep- tional object E in a triangulated category coincide with the perpendicular category of E. Based on this characterisation, we prove that the subcategory of fixed points of mutation in the derived category of the coherent sheaves on weighted projective line with genus one is equivalent to the derived category of a hereditary algebra. Meanwhile, we induce two new recollements by left and right mutations from a given recollement.

Lower Bounds for Local Cohomology Modules with Respect to a Pair of Ideals

Let R be a Noetherian ring, I and J two ideals of R, M an R-module and t an integer. Let S be a Serre subcategory of the category of R-modules satisfying the co ondition CI, and N be a finitely generated R-module with SuppRN= V（a） for some a ∈ W（I, J）. It is shown that if ExtR^J（N,HI^i,J（M）） ∈ S for all i 〈 t and all j 〈 t - i, then Ha^i（M） ∈ S for all i 〈 t. Let S be the class of all R-modules N with divmR N ≤ k, where k is an integer. It is proved that if Ha^i（M） ∈ S for all i 〈 t and all a ∈ W（I, J）, then HI^i,j（M） ∈ S for all i 〈 t. It follows that inf{i ： HI^i,j（M） S} = inf（inf{i ： Ha^i（M） S）： a ∈W（I,J）}.

Resolving Subcategories of Triangulated Categories and Relative Homological Dimension

We introduce and study （pre）resolving subcategories of a triangulated category and the homological dimension relative to these subcategories. We apply the obtained properties to relative Gorenstein categories.

Homological Dimensions Relative to Special Subcategories

Let A be an abelian category,C an additive,full and self-orthogonal subcategory of A closed under direct summands,rG(C)the right Gorenstein subcategory of A relative to C,and⊥C the left orthogonal class of C.For an object A in A,we prove that if A is in the right 1-orthogonal class of rG(C),then the C-projective and rG(C)-projective dimensions of A are identical;if the rG(C)-projective dimension of A is finite,then the rG(C)-projective and⊥C-projective dimensions of A are identical.We also prove that the supremum of the C-projective dimensions of objects with finite C-projective dimension and that of the rG(C)-projective dimensions of objects with finite rG(C)-projective dimension coincide.Then we apply these results to the category of modules.

On the monomorphism category of n-cluster tilting subcategories

Let M be an n-cluster tilting subcategory of mod-Λ,whereΛis an Artin algebra.Let S(M)denote the full subcategory of S(Λ),the submodule category of Λ,consisting of all the monomorphisms in M.We construct two functors from S(M)to mod-Λ,the category of finitely presented additive contravariant functors on the stable category of M.We show that these functors are full,dense and objective and hence provide equivalences between the quotient categories of S(M)and mod-Λ.We also compare these two functors and show that they differ by the n-th syzygy functor,provided M is an n Z-cluster tilting subcategory.These functors can be considered as higher versions of the two functors studied by Ringel and Zhang(2014)in the case Λ=k[x]/and generalized later by Eiríksson(2017)to self-injective Artin algebras.Several applications are provided.