THE TWO FUNCTORS IN THE FUZZY MODULAR CATEGORY

After defining Hom(chi (A), eta (B)) and chi (A) circle times eta (B) in the fuzzy modular category Fm, the sufficient conditions of the existence for exact Hom functors Hom(delta (M),), and Hom(, delta (M)), as well as exact Tensor functors delta (M)circle times and circle times delta (M) are given in this paper. Finally the weak isomorphisms relations between Horn functors and Tensor functors are displayed.

Gorenstein Projective Coresolutions and Co-Tate Homology Functors

For a local commutative Gorenstein ring R,Enochs et al.in[Gorenstein projective resolvents,Comm.Algebra 44(2016)3989-4000)defined a functor Extn^(R)(-,-)and showed that this functor can be computed by taking a totally acyclic complex arising from a projective coresolution of the first component or a totally acyclic complex arising from a projective resolution of the second component.In order to define the functor Extn^(R)(-,-)over general rings,we introduce the right Gorenstein projective dimension of an R-module M,RGpd(M),via Gorenstein projective coresolutions,and give some equivalent characterizations for the finiteness of RGpd(M).Then over a general ring R we define a co-Tate homology group Extn^(R)(-,-) for R-modules M and N with RGpd(M)<oo and Gpd(N)<∞,and prove that Extn^(R)(M,N)can be computed by complete projective coresolutions of the first variable or by complete projective resolutions of the second variable.

Relative Left Derived Functors of Tensor Product Functors

We introduce and study the relative lett derive functor Torn（￡,￡1） category, which unifies several related left derived functors. Then we give some criteria for computing the -resolution dimensions of modules in terms of the properties of Torn（￡,￡1） . We also construct a complete and hereditary cotorsion pair relative to balanced pairs. Some known results are obtained as corollaries.

Casacuberta's questions in extending localization functor

Let l denote a given category. A morphism f: A→B and an object X in l are said to be orthogonal if the function f~*:l(B, X)→l(A, X) is bijective. For a class of morphism S, S~⊥ denotes the class of objects which are orthogonal to each f∈S. Similarly,

Naturally Full Functors in Nature

We introduce and discuss the notion of a naturally full functor, The definition is similar to the definition of a separable functor; a naturally full functor is a functorial version of a full functor, while a separable functor is a functorial version of a faithful fimctor, We study the general properties of naturally full functors. We also discuss when functors between module categories and between categories of comodules over a coring are naturally full.

Adjoint Functors and Representation Dimensions

We study the global dimensions of the coherent functors over two categories that are linked by a pair of adjoint functors. This idea is then exploited to compare the representation dimensions of two algebras. In particular, we show that if an Artin algebra is switched from the other, then they have the same representation dimension.

Applications of BGP-reflection functors: isomorphisms of cluster algebras

Given a symmetrizable generalized Cartan matrix A,for any index k,one can define the matrix A (see sec.4 for the precise meaning).When A is of finite type,these isomorphisms behave nicely; they are compatible with the BGP-reflection functors of cluster categories defined in a previous work if we identify the indecomposable objects in the categories with cluster variables of the corresponding cluster algebras,and they are also compatible with the "truncated simple reflections" defined by Fomin-Zelevinsky.Using the construction of preprojective or preinjective modules of hereditary algebras by Dlab-Ringel and the Coxeter automorphisms (i.e.a product of these isomorphisms),we construct infinitely many cluster variables for cluster algebras of infinite type and all cluster variables for finite types.

(m,n)-内射和(m,n)-平坦函子

设R是有单位元的结合环,m和n是两个固定的正整数.引入了(m,n)-内射函子和(m,n)-平坦函子的概念,研究了这两类函子的一些性质,利用(m,n)-内射函子和(m,n)-平坦函子给出了(m,n)-凝聚环和Von Neumann正则环的一些等价刻画.

Frobenius双模和相对n-Gorenstein投射模

设R和S均是环,SMR是一Frobenius双模,MR是生成子.χ是一左R-模类,y是一左S-模类.在一定条件下,证明左R-模X是n-X-Gorenstein投射模当且仅当M■RX是n-y-Gorenstein投射左S-模.进而证明当F:RM→SM是忠实的Frobenius函子时,n-GX-pdR(X)=n-Gy-pdS(F(X)).

Distributive Functors

The distributivity of the forgetful functor F and of certain subfunctors of F has been studied.

Half Exact Functors Associated with Cotorsion Pairs on Exact Categories

In the previous article "Hearts of twin cotorsion pairs on exact categories. J. Algebra, 394,245–284(2013)", we introduced the notion of the heart for any cotorsion pair on an exact category with enough projectives and injectives, and showed that it is an abelian category. In this paper, we construct a half exact functor from the exact category to the heart. This is an analog of the construction of Abe and Nakaoka for triangulated categories. We will also use this half exact functor to find out a sufficient condition when two different hearts are equivalent.

Triangulated Structures Induced by Triangle Functors

Given a triangle functor F : A → B, the authors introduce the half image hIm F,which is an additive category closely related to F. If F is full or faithful, then hIm F admits a natural triangulated structure. However, in general, one can not expect that hIm F has a natural triangulated structure. The aim of this paper is to prove that hIm F admits a natural triangulated structure if and only if F satisfies the condition(SM). If this is the case, hIm F is triangle-equivalent to the Verdier quotient A/Ker F.

Schur Functors on QF-3 Standardly Stratified Algebras

Let A be a QF-3 standardly stratified algebra and f be a Schur functor corresponding to some projective-injective faithful A-module, denoted by Ae. The main result of this paper is to prove that, if the dominant dimension of A is sufficiently large, then ] induces a full embedding from ￡（△） to eAe-mod which preserves Ext-groups up to certain degrees, where ￡（△） denotes the full subcategory of A-mod whose objects are filtered by standard A-modules. We check this criterion on some typical examples, quantized Schur algebras Sq（n,r） with n≥r and finite-dimensional algebras associated with the Bernstein-Gelfand-Gelfand category O of semisimple complex Lie algebras.

Functors of the Category of Abelian Sheaves on Regular Affine Scheme

In this short paper, we prove that if R is a regular local ring of unequal characteristic, then there exists an additive covariant functor G from the category of abelian sheaves on SpecR to the category of abelian groups such that id_R(G(R))】dimG(R). This result shows that the answer to the question 3.8 (ii) in [3] may be negative.

Some Pairs of Adjoint Functors Involving Track Homotopy Categories

K. A. Hardie and K. H. Kamps investigated the track homotopy category H_B over a fixed space B ([1]). They have introduced two pairs of adjoint functors: P_B -|N_B and m_* -| m~*, where P_B:H_B→H^B, and m_*:H_A→H_B for a fixed map m: A→B. We have introduced a split fibration of categories L: H_b→H_B and proved L-|J, J-|L in [2]. This paper first extends P_B-|N_B to P_b_*-|N_Bb~# for any fixed map b:B→.Moreover we also extend these results to obtain two pairs of adjoint functors involving track homotopy categories H_b and H^b where H^b is the dual of H_b. One of our results is N_b-|P_b. This differs from P_B-|N_B.

BGP-Reflection Functors and Lusztig's Symmetries of Modified Quantized Enveloping Algebras

Let U be a quantized enveloping algebra and U its modified form. Lusztig gives some symmetries on U and U. In view of the realization of U by the reduced Drinfeld double of the Ringel- Hall algebra, one can apply the BGP-refiection functors to the double Ringel-HM1 algebra to obtain Lusztig＇s symmetries on U and their important properties, for instance, the braid relations. In this paper, we define a modified form Hof the Ringel-Hall algebra and realize the Lusztig＇s symmetries on U by applying the BGP-reflection functors to H

A New Functor from D_5-Mod to E_6-Mod

We find a new representation of the simple Lie algebra of type E6 on the polynomial algebra in 16 variables, which gives a fractional representation of the corresponding Lie group on 16-dimensional space. Using this representation and Shen＇s idea of mixed product, we construct a new functor from Ds-Mod to E6-Mod. A condition for the functor to map a finite-dimensional irreducible Ds-module to an infinite-dimensional irreducible E6-module is obtained. Our results yield explicit constructions of certain infinite-dimensional irreducible weight E6-modules with finite-dimensional weight subspaces. In our approach, the idea of Kostant＇s characteristic identities plays a key role.

Objective Triangle Functors in Adjoint Pairs

An additive functor F： A→B between additive categories is objective if any morphism f in A with F（f） = 0 factors through an object K with F（K） = 0. We consider when a triangle functor in an adjoint pair is objective. We show that a triangle functor is objective provided that its adjoint （whatever left adjoint or right adjoint） is full or dense. We Mso give an example to show that the adjoint of a faithful triangle functor is not necessarily objective. In particular, the adjoint of an objective triangle functor is not necessarily objective. This is in contrast to the well-known fact that the adjoint of a triangle functor is always a triangle functor. Also, for an arbitrary a^tjoint pair （F, G） between categories which are not necessarily additive, we give a sufficient and necessary condition such that F （resp., G） is full or faithful.

The Cotorsion Pair Cogenerated by the Subcategory of Stable Functors in ((R-mod)^op, Ab)

Let R be an associative ring with identity. Denote by ((R-mod)^op, Ab) the category consisting of contravariant functors from the category of finitely presented left R-modules R-mod to the category of abelian groups Ab. An object in ((R-mod)^op, Ab) is said to be a stable functor if it vanishes on the regular module R. Let T be the subcategory of stable functors. There are two torsion pairs t1=(Gen(-,R),T)and t2=(T,F1), where 1 is the subcategory of ((R-mod)^op, Ab) consisting of functors with flat dimension at most 1. In this article, let R be a ring of weakly global dimension at most 1, and assume R satisfies that for any exact sequence 0 → M → N → K → 0, if M and N are pure injective, then K is also pure injective. We calculate the cotorsion pair (⊥T,(⊥T)⊥)cogenerated by T clearly. It is shown that G∈⊥T if and only if G/t1(G) is a projective object in T, i.e., G/t1(G)=(-,M) for some R-module M;and G∈(⊥T)⊥ if and only if G/t2(G) is of the form (-, E), where E is an injective R-module.

On subgroup functors of finite soluble groups

The principal aim of this paper is to study the regular and transitive subgroup functors in the universe of all finite soluble groups.We prove that they form a complemented and non-modular lattice containing two relevant sublattices.This is the answer to a question(Question 1.2.12)proposed by Skiba(1997)in the finite soluble universe.