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.

Applications of BGP-reflection functors: isomorphisms of cluster algebras

Given a symmetrizable generalized Cartan matrix A, for any index k, one can define an automorphism associated with A, of the field Q(u1,…, un) of rational functions of n independent indeterminates u1,…,un.It is an isomorphism between two cluster algebras associated to 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 DIab-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.

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.

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.

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

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.

Distributive Functors

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

Objective triangle functors

An additive functor F:A→B between additive categories is said to be objective,provided any morphism f in A with F(f)=0 factors through an object K with F(K)=0.We concentrate on triangle functors between triangulated categories.The first aim of this paper is to characterize objective triangle functors F in several ways.Second,we are interested in the corresponding Verdier quotient functors VF:A→A/Ker F,in particular we want to know under what conditions VF is full.The third question to be considered concerns the possibility to factorize a given triangle functor F=F2F1with F1a full and dense triangle functor and F2a faithful triangle functor.It turns out that the behavior of splitting monomorphisms and splitting epimorphisms plays a decisive role.

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.

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.

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.

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.

仿函数配接器的扩展和实现

C++标准库的STL中用到了仿函数(也称为函数对象)的概念,实际上,其中所有的算法都提供了一个用仿函数作为其参数的重载版本,从而决定其行为。对于每个普通的一元和二元运算符,标准库都提供了一个仿函数,对于组合而言,标准库还提供了几个标准的仿函数和配接器,然而,C++标准库并没有提供足够的组合型配接器,用于支持仿函数之间的组合。本文将对一些常用的仿函数组合进行扩展,并具体实现出几个自定义仿函数和几个常用的组合型配接器。

On Ding homological dimensions

In this paper, we investigate Ding projective dimensions and Ding injective di- mensions of modules and rings. Let R be a ring with rDPD（R） = n 〈 ∞, and let YYl = {Mild（M） 〈 ∞}. We prove that （DP,W1） is a complete hereditary cotorsion pair such that a module M belongs to DD∩W1 if and only if M is projective, moreover, 1421 = （M[pd（M） 〈 ∞} = {MIfd（M） ≤ n} = {MIpd（M） ≤ n}. Then we introduce and inves- tigate Ding derived functor Dext^i（-, -）, and use it to characterize global Ding dimension. We show that if R is a Ding-Chen ring, or if R is a ring with rDPD（R） 〈≤ and rDID（R） 〈 ≤, then rDPD（R） 〈 n if and only if rDID（R） 〈 n if and only if Dext^n＋i（M,N） = 0 for all modules M and N and all integer i ≥ 1.

On a Generalized Matrix Algebra over Frobenius Algebra

Let A be a Frobenius k-algebra. The matrix algebra R =(■) is called a generalized matrix algebra over a Frobenius algebra A. In this paper we show that R is also a Frobenius algebra.