二面体群的Grothendieck环结构

二面体群的表示范畴为对称半单monoidal范畴,因而其Grothendieck环为有限多个元素生成的交换环.本文确定了该Grothendieck环的极小生成元,并且进一步证明了该Grothendieck环与某一多项式环的商环同构.     

二面体群的Grothendieck环结构

二面体群的表示范畴为对称半单monoidal范畴,因而其Grothendieck环为有限多个元素生成的交换环.本文确定了该Grothendieck环的极小生成元,并且进一步证明了该Grothendieck环与某一多项式环的商环同构.     

有限生成亚投射模范畴

对一个QF环R,本文证明:其投射左R模范畴是因式分解范畴当且仅当gl.dim R≤1．进一步,若 P(_RR)＝P(R_R)＝0,则其通过左模而得到的亚 Crothendieck群与其通过右模而得到的亚Grothendieck群在同构意义下是一样的．还证明了有限生成亚投射左R－模范畴不仅是一个因式分解范畴而且是一个带积的具有小的骨架子范畴的范畴．     

n-角范畴的局部化

本文研究了n-角范畴的局部化理论.利用给定的n-角范畴K和K的一个相容乘法系S,通过局部化方法构造与原范畴对象相同的商范畴S-1K,在新范畴中S的态射成为同构,并且该商范畴具有n-角结构且满足一定的泛性,推广了三角范畴的局部化理论.     

左三角范畴的局部化

本文通过左挠对给出左三角范畴的局部化,证明了左三角范畴的局部化范畴仍然是左三角范畴,并证明了左三角范畴的正合列通过稳定化函子可以导出三角范畴正合列.     

L-fuzzy集理论的范畴基础与层表示

鉴于L-fuzzy集在理论上的重要性和应用上的广泛性,旨在建立L-fuzzy集理论的范畴基础与它的层表示,提出完备范畴中对象上的格值结构概念,这一概念是L-fuzzy结构在范畴层面上的提升,进一步提出完备范畴上格值结构提升范畴概念,证明了在集合范畴中L-fuzzy结构与格值结构是同构的.以集层、群层、环层和左R-模层以及Grothendieck层等概念为基础,提出完备范畴中对象上的层结构以及完备范畴上层结构提升范畴概念,证明了在集合范畴中L-fuzzy结构与层结构也是同构的.     

Topos理论简介

Topos概念的提出(即elementary topos的公理化)是在研究了如下几类范畴之后:Grothendieck topos,集合的范畴S,以及集合论的布尔值模型。 Grothendieck提出用拓扑空间X上的set valued sheaf的全体做成的范畴Sh(X)作为推广了的拓补空间X,用以研究空间X上的上同调.他把拓扑的概念推广到小范畴(smallcategory)C上,称为一个site(或称为Grothendieck拓扑,参看第2节)。这样就可以考虑给出的site(C,J)上的set valued sheaf的全体做成的范畴Sh(C,J),用它来研究site(C,J)     

定义在范畴_RM_n~l上的张量函数

文[1]引进了左R-n模范畴_RM_n~1,将Hom函子推扩到_RM_n~l上,且讨论了它的拟正合性。在模范畴中,张量函子与Hom函子一样是一个重要的函子。文[2]企图在_RM_n~l上建立相应的张量积,并证明张量函子M_n~:AG_(rn)→_RM_n~l与Hom函子Hom(M_):AG_(rn)′→_RM_n~l为伴随对。但其构造的交换自由n-群仅对满足(E)条件的交换n-群具有泛性,故其张量积亦只是     

具有左R-模结构的类型及其范畴逻辑模型

以范畴逻辑与类型论为基础,引入类型中的交换群理论、环理论以及左R-模理论.证明了类型中的交换群理论在满足分配律的范畴中的模型是交换群对象,环理论的模型是环对象,左R-模理论的模型是左R-模对象,并给出左R-模理论在集合范畴和层范畴等几个具体范畴中的模型.     

从Frobenius特征标公式到顶点算子

本文利用群表示论方法简洁证明了Frobenius对称群特征标公式和Specht圈群特征标公式. Frobenius关于对称群的特征标公式已经显现了顶点算子的雏形,本文提出的证明关键是利用对称群Grothendieck环的乘法公式得到Bernstein顶点算子.同时,本文将此方法推广到圈群链的Grothendieck环,利用顶点算子和广义Clifford代数导出圈群的不可约特征标,简单地证明了Specht关于圈群的特征标公式.     

Idempotent Completion of n-Angulated Categories

Applied Categorical Structures - Let $${\mathcal {C}}$$ be an n-angulated category. We prove that its idempotent completion $$\widetilde{{\mathcal {C}}}$$ admits a unique n-angulated structure such...     

Locally Stable Grothendieck Categories: Applications

We introduce for any Grothendieck category the notion of stable localizing subcategory, as a localizing subcategory that can be written as an intersection of localizing subcategories defined by indecomposable injectives. A Grothendieck category in which every localizing subcategory is stable is called a locally stable category. As a main result we give a characterization of these categories in terms of the local stability of a localizing subcategory and its quotient category. The locally coirreducible categories (in particular, the categories with Gabriel dimension) and the locally noetherian categories are examples of locally stable categories. We also present some applications to the category of modules over a left fully bounded noetherian ring, to the category of comodules over a coalgebra and to the category of modules over graded rings.     

Precosheaves of pro-sets and abelian pro-groups are smooth

Let D be the category of pro-sets (or abelian pro-groups). It is proved that for any Grothendieck site X, there exists a reflector from the category of precosheaves on X with values in D to the full subcategory of cosheaves. In the case of precosheaves on topological spaces, it is proved that any precosheaf is smooth, i.e. is locally isomorphic to a cosheaf. Constant cosheaves are constructed, and there are established connections with shape theory.     

The Grothendieck Group of an n-exangulated Category

Applied Categorical Structures - We define the Grothendieck group of an n-exangulated category. For n odd, we show that this group shares many properties with the Grothendieck group of an exact or...     

The Lagrangian cobordism group of $$T^2$$

We compute the Lagrangian cobordism group of the standard symplectic 2-torus and show that it is isomorphic to the Grothendieck group of its derived Fukaya category. The proofs use homological mirror symmetry for the 2-torus.     

The Ziegler Spectrum of a Locally Coherent Grothendieck Category

The general theory of locally coherent Grothendieck categoriesis presented. To each locally coherent Grothendieck categoryC a topological space, the Ziegler spectrum of C, is associated.It is proved that the open subsets of the Ziegler spectrum ofC are in bijective correspondence with the Serre subcategoriesof coh C the subcategory of coherent objects of C. This is aNullstellensatz for locally coherent Grothendieck categories.If R is a ring, there is a canonical locally coherent Grothendieckcategory RC (respectively, CR) used for the study of left (respectively,right) R-modules. This category contains the category of R-modulesand its Ziegler spectrum is quasi-compact, a property used toconstruct large (not finitely generated) indecomposable modulesover an artin algebra. Two kinds of examples of locally coherentGrothendieck categories are given: the abstract category theoreticexamples arising from torsion and localization and the examplesthat arise from particular modules over the ring R. The dualitybetween coh-(RC) and coh-CR is shown to give an isomorphismbetween the topologies of the left and right Ziegler spectraof a ring R. The Nullstellensatz is used to give a proof ofthe result of Crawley-Boevey that every character : K₀(coh-C) Z is uniquely expressible as a Z-linear combination of irreduciblecharacters. 1991 Mathematics Subject Classification: 16D90,18E15.     

The Exchange Property in Grothendieck Categories: Applications

We introduce the concept of an object with the (finite) exchange property in an arbitrary Grothendieck category, and we present the basic properties of such an object. Applications are given for categories of graded modules and for categories of comodules over a coalgebra. Among other results, it is proved that an arbitrary coalgebra 𝒞 over a field has the finite exchange property.