First,Second, and Third Change of Rings Theorems for Gorenstein Homological dimensions

In this article, we investigate the change of rings theorems for the Gorenstein dimensions over arbitrary rings. Namely, by the use of the notion of strongly Gorenstein modules, we extend the well-known first, second, and third change of rings theorems for the classical projective and injective dimensions to the Gorenstein projective and injective dimensions, respectively. Each of the results established in this article for the Gorenstein projective dimension is a generalization of a G-dimension of a finitely generated module M over a noetherian ring R.     

Weyl代数的表示

设Ｋ是一个域．证明了，若ｃｈＫ＝０，那么ｎ－ｔｈＷｅｙｌ代数Ａ_ｎ（ｋ）没有有限维表示．还给出了Ａ_ｎ（ｋ）的不可约Ｈａｒｉｓｈ－Ｃｈａｎｄｒａ模的分类．当Ｋ是一个特征非零的代数闭域时，给出了有限维不可约Ａ_ｎ（Ｋ）－模的分类．     

Hopf代数交叉积的同调维数

Let H be a finite dimensional cocommutative Hopf algebra and A an H-module algebra, ln this paper, we characterize the projectivity (injectivity) of M as a left A#σ H-module when it is projective (injective) as a left A-module. The sufficient and necessary condition for A#σ H, the crossed product, to have finite global homological dimension is given, in terms of the global homological dimension of A and the surjectivity of trace maps, provided that H is cocommutative and A is commutative.     

Some Groups of Finite Homological Type

For each n≥ 0 we construct a torsion-free group that satisfies K.S. Brown’s FHT condition and is F _n (and hence FP _n ), but is not FP _n+1.I. J. Leary was partially supported by NSF grant DMS-0505471.M. Saadetoğlu was supported by the British Council and by the Ohio State Mathematical Research Institute     

代数的张量积的同调满同态的构造

利用已知的代数的同调满同态来构造其张量积代数的同调满同态.设A,B,C,D是域k上的有限维代数,如果环同态f:A→C和g:B→D是环的同调满同态,则fg:AB→CD也是环的同调满同态.     

Homological Properties of Modules Over Group Algebras

Let G be a locally compact group, and let L¹ (G) be the Banachalgebra which is the group algebra of G. We consider a varietyof Banach left L¹ (G)-modules over L¹ (G), and seek to determineconditions on G that determine when these modules are eitherprojective or injective or flat in the category. The answerstypically involve G being compact or discrete or amenable. Forexample, in the case where G is discrete and 1 < p < ,we find that the module ^p (G) is injective whenever G is amenable,and that, if it is amenable, then G is ‘pseudo-amenable’,a property very close to that of amenability. 2000 MathematicsSubject Classification 46H25, 43A20.     

非有限阶化无中心Virasoro超代数的表示

介绍非有限阶化无中心Virasoro超代数的概念,并通过具体计算给出了非有限阶化无中心Virasoro超代数的一类中间序列模.     

关于标准分层代数的多项式代数的滤链维数

标准分层代数是拟遗传代数的推广,其性质和理论意义受到人们的重视.在本文中,设A是域k上的标准分层代数,我们从特征模的角度,对A的多项式代数A[x]上的滤链维数进行了研究,得到了一些有意义的结果.     

有限复杂度的Koszul代数

首先给出了Koszul代数的张量积的复杂度,然后研究了Koszul遗传代数上的Koszul单列模,并证明了Koszul遗传代数上的Koszul模M的Koszul合成列在同构意义下是唯一的.     

Homological and finiteness properties of picture groups

Picture groups are a class of groups introduced by Guba and Sapir. Known examples include Thompson's groups , , and .

In this paper, a large class of picture groups is proved to be of type . A Morse-theoretic argument shows that, for a given picture group, the rational homology vanishes in almost all dimensions.

     

Representations of Ariki-Koike Algebras and Grobner-Shirshov Bases

In this paper, we investigate the structure of Ariki–Koikealgebras and their Specht modules using Gröbner–Shirshovbasis theory and combinatorics of Young tableaux. For a multipartition, we find a presentation of the Specht module S given by generatorsand relations, and determine its Gröbner–Shirshovpair. As a consequence, we obtain a linear basis of S consistingof standard monomials with respect to the Gröbner–Shirshovpair. We show that this monomial basis can be canonically identifiedwith the set of cozy tableaux of shape . 2000 Mathematics SubjectClassification 16Gxx, 05Exx.     

相对Hopf模的同调维数与半单性

Let H be a Hopf algebra over a field with bijective antipode, and A a commutative cleft right H-comodule algebra. In this paper, we investigate the ho-mological dimensions and the semisimplicity of the category of relative Hopf modulesAMH.     

A Homological Approach to Two Problems on Finite Sets

We propose a homological approach to two conjectures descended from the Erdös-Ko-Rado Theorem, one due to Chvátal and the other to Frankl and Füredi. We apply the method to reprove, and in one case improve, results of these authors related to their conjectures.     

群与代数的表示理论（迎接ICM2002特约文章）

有限群模表示论,代数群与量子群,代数表示论,同调代数与代数K-理论是当前国际数学研究的前沿重点领域,在20世纪得到巨大发展,它们之间的相互交叉融合与综合统一在数学发展中具有鲜明特色。本文简要概述上述研究领域的发展历史与最新研究进展。     

On the Existence of Certain Modules of Finite Gorenstein Homological Dimensions

Let (R, 𝔪) be a commutative Noetherian local ring. It is known that R is Cohen–Macaulay if there exists either a nonzero finitely generated R-module of finite injective dimension or a nonzero Cohen–Macaulay R-module of finite projective dimension. In this article, we investigate the Gorenstein analogues of these facts.     

Rational and Homological Equivalence of Real Algebraic Cycles

In this paper we study algebraic cycles on smooth quasiprojective real algebraic varieties. We are interested under what conditions an algebraic cycle homologous to zero is rationally equivalent to a cycle supporting no real points.     

A Note on Periodicity in Representation Theory of Finite Dimensional Algebras

Let Λ be an algebra over an algebraically closed field K of infinite transcendence degree over its prime subfield. We prove in particular that if every s-dimensional indecomposable nonprojective Λ-module is Ω-periodic (resp. DTr-periodic) then there exists a common bound for the Ω-periods (resp. DTr-periods) of s-dimensional indecomposable nonprojective Λ-modules.     

Homological Dimension of Coalgebras and Crossed Coproducts

Let H be a Hopf k-algebra. We study the global homological dimension of the underlying coalgebra structure of H. We show that gl.dim(H) is equal to the injective dimension of the trivial right H-comodule k. We also prove that if D = C H is a crossed coproduct with invertible , then gl.dim(D) gl.dim(C) + gl.dim(H). Some applications of this result are obtained. Moreover, if C is a cocommutative coalgebra such that C ^* is noetherian, then the global dimension of the coalgebra C coincides with the global dimension of the algebra C ^*.     

Relative Projectivity of Modules and Cohomology Theory of Finite Groups

In the modular representation theory of finite groups the theory of projectivity relative to subgroups is of fundamental importance. To generalize this notion the theory of projectivity relative to modules was introduced by the first author. Our aim is to show some aspects of cohomology theory of finite groups concerning projectivity of modules relative to both subgroups and modules. We shall give some applications to the cohomology theory; especially we shall calculate the mod 2 cohomology algebras of finite groups with wreathed Sylow 2-subgroups.