Tilting and cluster tilting for quotient singularities

We shall show that the stable categories of graded Cohen–Macaulay modules over quotient singularities have tilting objects. In particular, these categories are triangle equivalent to derived categories of finite dimensional algebras. Our method is based on higher dimensional Auslander–Reiten theory, which gives cluster tilting objects in the stable categories of (ungraded) Cohen–Macaulay modules.     

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.     

COPURE INJECTIVE MODULES

Abstract

A module is said to be copure injective if it is injective with respect to all modules A ? B with B/A injective. We first characterize submodules that have the extension property with respect to copure injective modules. Then we characterize commutative rings with finite self injective dimension in terms of copure injective modules. Finally, we show that the quotient categories of reduced copure injective modules and reduced h- divisible modules are isomorphic.     

Essentially normal Hilbert modules and Khomology Ⅲ: Homogenous quotient modules of Hardy modules on the bidisk

In this paper, we study the homogenous quotient modules of the Hardy module on the bidisk. The essential normality of the homogenous quotient modules is completely characterized. We also describe the essential spectrum for a general quotient module. The paper also considers K-homology invariant defined in the case of the homogenous quotient modules on the bidisk.     

Essentially normal Hilbert modules and K-homology Ⅲ:Homogenous quotient modules of Hardy modules on the bidisk

In this paper, we study the homogenous quotient modules of the Hardy module on the bidisk. The essential normality of the homogenous quotient modules is completely characterized. We also describe the essential spectrum for a general quotient module. The paper also considers K-homology invariant defined in the case of the homogenous quotient modules on the bidisk.     

Essentially normal Hilbert modules and K-homology IV: Quasi-homogenous quotient modules of Hardy module on the polydisks

In this paper,we give a complete characterization for the essential normality of quasi-homogenous quotient modules of the Hardy modules H2 (D2).Also,we show that if d 3,then all the principle homogenous quotient modules of H 2 (Dd) are not essentially normal.     

On Ext-Transfer for Algebraic Groups

This paper builds upon the work of Cline and Donkin to describe explicit equivalences between some categories associated to the category of rational modules for a reductive group G and categories associated to the category of rational modules for a Levi subgroup H. As an application, we establish an Ext-transfer result from rational G-modules to rational H-modules. In case G = GL_n, these results can be illustrated in terms of classical Schur algebras. In that case, we establish another category equivalence, this time between the module category for a Schur algebra and the module category for a union of blocks for a natural quotient of a larger Schur algebra. This category equivalence provides a further Ext-transfer theorem from the original Schur algebra to the larger Schur algebra. This result extends to the category level the decomposition number method of Erdmann. Finally, we indicate (largely without proof) some natural variations to situations involving quantum groups and q-Schur algebras.     

Essentially normal Hilbert modules and Khomology Ⅲ: Homogenous quotient modules of Hardy modules on the bidisk

In this paper, we study the homogenous quotient modules of the Hardy module on the bidisk. The essential normality of the homogenous quotient modules is completely characterized. We also describe the essential spectrum for a general quotient module. The paper also considers K-homology invariant defined in the case of the homogenous quotient modules on the bidisk. This work is partially supported by the National Natural Science Foundation of China (Grant No. 10525106), the Young Teacher Fund, the National Key Basic Research Project of China (Grant No. 2006CB805905) and the Specialized Research for the Doctoral Program     

Quotient modules for some Hilbert modules over the bidisk

In this note, we will prove the essential normality of some type of quotient Hilbert modules over the bidisk. Moreover, we will study some properties of the compression of the multiplication operators to the quotient Hilbert modules.     

Quasi-Whittaker modules over the conformal Galilei algebras

In this paper, we study quasi-Whittaker modules over the conformal Galilei algebras. We determine all quasi-Whittaker vectors and the simplicities of the universal quasi-Whittaker modules. For the reducible ones, we determine the corresponding simple quotient modules. Thus we classify all simple quasi-Whittaker modules for the conformal Galilei algebras.     

Essential normality of quasi-homogeneous quotient modules over the polydisc

In this paper, we characterize the essential normality of quasi-homogeneous quotient Hardy modules over the polydisc by giving a complete criterion in terms of partially maximal ideals. Then we generalize this result to the weighted Bergman modules.     

Beurling type quotient modules over the bidisk and boundary representations

This paper mainly concerns Beurling type quotient modules of H²(D²) over the bidisk. By establishing a theorem of function theory over the bidisk, it is shown that a Beurling type quotient module is essentially normal if and only if the corresponding inner function is a rational inner function having degree at most (1,1). Furthermore, we apply this result to the study of boundary representations of Toeplitz algebras over quotient modules. It is proved that the identity representation of C^∗(Sz,Sw) is a boundary representation of B(Sz,Sw) in all nontrivial cases. This extends a result of Arveson to Toeplitz algebras on Beurling type quotient modules over the bidisk (cf. [W. Arveson, Subalgebras of C^∗-algebras, Acta Math. 123 (1969) 141-224; W. Arveson, Subalgebras of C^∗-algebras II, Acta Math. 128 (1972) 271-308]). The paper also establishes K-homology defined by Beurling type quotient modules over the bidisk.     

Curvature invariant and generalized canonical operator models – II

In Douglas et al. (2012) [9], the authors investigated a family of quotient Hilbert modules in the Cowen–Douglas class over the unit disk constructed from classical Hilbert modules such as the Hardy and Bergman modules. In this paper we extend the results to the multivariable case of higher multiplicity. Moreover, similarity as well as isomorphism results are obtained.     

Star-generating vectors of Rudin's quotient modules

The purpose of this paper is to study a class of quotient modules of the Hardy module H²(Dⁿ)$H^2({\mathbb{D}}^n)$. Along with the two variables quotient modules introduced by W. Rudin, we introduce and study a large class of quotient modules, namely Rudin's quotient modules of H²(Dⁿ)$H^2({\mathbb{D}}^n)$. By exploiting the structure of minimal representations we obtain an explicit co-rank formula for Rudin's quotient modules.     

A generalization of the category O of Bernstein–Gelfand–Gelfand

In the study of simple modules over a simple complex Lie algebra, Bernstein, Gelfand and Gelfand introduced a category of modules which provides a natural setting for highest weight modules. In this note, we define a family of categories which generalizes the BGG category. We classify the simple modules for some of these categories. As a consequence we show that these categories are semisimple.     

A note on strong protomodularity,actions and quotients

In order to study the problems of extending an action along a quotient of the acted object and along a quotient of the acting object, we investigate some properties of the fibration of points. In fact, we obtain a characterization of protomodular categories among quasi-pointed regular ones, and, in the semi-abelian case, a characterization of strong protomodular categories. Eventually, we return to the initial questions by stating the results in terms of internal actions.     

量子环面C-1上的一族Noether模

记C_-1为q=-1 的量子环面Lie代数. 本文以Laurent 多项式环C[x^±1] 为表示空间, 构造了C_-1上的一类Noether 但非Artin 的模, 并确定了它们的全部子模和商模以及它们之间的全体模同态,最后指出该模与A₁型loop代数忠实表示的关系.     

Monoidal Entwined模范畴上的张量积恒等式

本文研究了monoidal entwined模范畴上的张量积恒等式.利用了monoidal entwined模范畴的性质及Doi-Hopf模范畴上的张量积恒等式的研究方法,获得了monoidal entwined模范畴上的一些张量积恒等式,并证明了entwined模范畴有足够的内射对象,结果推广了Doi-Hopf模范畴的结论.     

Strongly W-Gorenstein modules

Let W be a self-orthogonal class of left R-modules. We introduce a class of modules, which is called strongly W-Gorenstein modules, and give some equivalent characterizations of them. Many important classes of modules are included in these modules. It is proved that the class of strongly W-Gorenstein modules is closed under finite direct sums. We also give some sufficient conditions under which the property of strongly W-Gorenstein module can be inherited by its submodules and quotient modules. As applications, many known results are generalized.