Tilting subcategories in extriangulated categories

Extriangulated category was introduced by H.Nakaoka and Y.Palu to give a unification of properties in exact categories anjd triangulated categories.A notion of tilting(resp.,cotilting)subcategories in an extriangulated category is defined in this paper.We give a Bazzoni characterization of tilting(resp.,cotilting)subcategories and obtain an Auslander-Reiten correspondence between tilting(resp.,cotilting)subcategories and coresolving covariantly(resp.,resolving contravariantly)finite subcatgories which are closed under direct summands and satisfy some cogenerating(resp.,generating)conditions.Applications of the results are given:we show that tilting(resp.,cotilting)subcategories defined here unify many previous works about tilting modules(subcategories)in module categories of Artin algebras and in abelian categories admitting a cotorsion triples;we also show that the results work for the triangulated categories with a proper class of triangles introduced by A.Beligiannis.     

关于余倾斜双模的k-挠自由模

Let A and F be left and right Noetherian rings and ∧ωr a cotilting bimodule. A necessary and sufficient condition for a finitely generated A-module to be ω-k-torsionfree is given and the extension closure of Tω^i is discussed. As applications, we give some results of ∧ωr related to l.id（ω） ≤ k.     

Relative syzygies and grade of modules

Recently, Takahashi established a new approximation theory for finitely generated modules over commutative Noetherian rings, which unifies the spherical approximation theorem due to Auslander and Bridger and the Cohen-Macaulay approximation theorem due to Auslander and Buchweitz. In this paper we generalize these results to much more general case over non-commutative rings. As an application, we establish a relation between the injective dimension of a generalized tilting module ω and the finitistic dimension with respect to ω.     

对偶的Hom型拟Hopf代数的余表示和范畴实现

In this paper, we introduce the dual Hom-quasi-Hopf algebra and prove that the comodules category of a(braided) dual Hom-quasi-bialgebra is a monoidal category. Finally,we give a categorical realization of dual Hom-quasi-Hopf algebras.     

Equivalences between categories of modules and categories of comodules

We show the close connection between apparently different Galois theories for comodules introduced recently in [J. Gomez-Torrecillas and J. Vercruysse, Comatrix corings and Galois Comodules over firm rings, Algebr. Represent. Theory, 10 （2007）, 271 306] and [Wisbauer, On Galois comodules, Comm. Algebra 34 （2006）, 2683-2711]. Furthermore we study equivalences between categories of comodules over a coring and modules over a firm ring. We show that these equivalences are related to Galois theory for comodules.     

幺半群上的斜强可逆环

For a monoid M, we introduce the concept of skew strongly M-reversible rings which is a generalization of strongly M-reversible rings, and investigate their properties. It is shown that if G is a finitely generated Abelian group, then G is torsion-free if and only if there exists a ring R with |R| ≥ 2 such that R is skew strongly G-reversible. Moreover, we prove that if R is a right Ore ring with classical right quotient ring Q, then R is skew strongly M-reversible if and only if Q is skew strongly M-reversible.     

Tilting objects on tubular weighted projective lines: A cluster tilting approach

Using the cluster tilting theory,we investigate the tilting objects in the stable category of vector bundles on a weighted projective line of weight type(2,2,2,2).More precisely,a tilting object consisting of rank-two bundles is constructed via the cluster tilting mutation.Moreover,the cluster tilting approach also provides a new method to classify the endomorphism algebras of the tilting objects in the category of coherent sheaves and the associated bounded derived category.     

POINCAR SERIES AND AN APPLICATION TO WEYL ALGEBRAS

Let A n be the n-th Weyl algebra over a field of characteristic 0 and M a finitely generated module over A n . By further exploring the relationship between the Poincar′e series and the dimension and the multiplicity of M , we are able to prove that the tensor product of two finitely generated modules over A n has the multiplicity equal to the product of the multiplicities of both modules. It turns out that we can compute the dimensions and the multiplicities of some homogeneous subquotient modules of A n .     

Poincaré series and an application to Weyl algebras

Let A n be the n-th Weyl algebra over a field of characteristic 0 and M a finitely generated module over A n . By further exploring the relationship between the Poincar′e series and the dimension and the multiplicity of M , we are able to prove that the tensor product of two finitely generated modules over A n has the multiplicity equal to the product of the multiplicities of both modules. It turns out that we can compute the dimensions and the multiplicities of some homogeneous subquotient modules of A n .     

Injective Cogenerators,Cotilting Modules and Cosilting Modules

It is well known that the concept of cotilting modules generalizes injective cogenerators and in turn,the concept of cosilting modules generalizes cotilting modules.In this paper,we further investigate the close connections among injective cogenerators,cotilting modules and cosilting modules from the viewpoint of morphism categories.Some applications are also given.     

Gorenstein Cotilting and Tilting Modules

Auslander and Solberg introduced the concepts of finitely generated cotilting and tilting modules in relative homological algebra considering subfunctors of the Ext-functor. In this article we generalize Auslander–Solberg relative notions by giving the definitions of infinitely generated Gorenstein cotilting and tilting modules by means of Gorenstein exact sequences. Using the theory developed by Enochs on the existence of Gorenstein preenvelopes and precovers, we prove a characterization of relative Gorenstein cotilting and tilting modules, which is a generalization of the beautiful characterization of relative cotilting and tilting modules given by Bazzoni.     

有限内射维数的余倾斜余模(英文)

本文中,受C.Nastasescu etc.和Y.Miyashita思想的影响,定义了余代数的余倾斜余模,研究得出有限内射维数的余倾斜余模的一些结论.     

Tilting modules over almost perfect domains

We provide a complete classification of all tilting modules and tilting classes over almost perfect domains, which generalizes the classifications of tilting modules and tilting classes over Dedekind and 1-Gorenstein domains. Assuming the APD is Noetherian, a complete classification of all cotilting modules is obtained (as duals of the tilting ones).     

Finitistic n-Self-Cotilting Modules

We study a class of modules which can be characterized using a duality theorem, called finitistic n-self-cotilting. Such a module Q can be characterized using dual conditions of some generalizations for star modules: every module M which has a right resolution with n terms isomorphic to finite powers of Q (i.e., M is n-finitely Q-copresented) has a right resolution with (n + 1) terms, and the functor Hom _R (?, Q) preserves the exactness of all monomorphisms with their ranges finite powers of Q and cokernels n-finitely Q-copresented modules. In the general case, these modules are independent toward other kinds of modules which are characterized using some dualities (w-Π _f -quasi injective modules, costar modules, f-cotilting modules). Closure properties for the classes involved in the duality are studied. At the end of the article, connections with the cotilting theory are exhibited, in the case of finitely dimensional algebras over fields.     

Partial cotilting modules and the lattices induced by them

We study a duality between (infinitely generated) cotilting and tilting modules over an arbitrary ring. Dualizing a result of Bongartz, we show that a module P is partial cotilting iff P is a direct summand of a cotilting module C such that the left Ext-orthogonal class ⊥P coincides with ⊥C. As an application, we characterize all cotilting torsion-free classes. Each partial cotilting module P defines a lattice L = [Cogen P₁P] of torsion-free classes. Similarly, each partial tilting module P′ defines a lattice L′ = [[Gen P′,P′⊥]] of torsion classes. Generalizing a result of Assem and Kerner, we show that the elements of L are determined by their Rej_p-torsion parts, and the elements of L′ by their Tr_p-torsion-free parts.     

关于内射维数有限的cotilting余模

令K是域,设C是一个K-余代数,T为cotilting左C-余模,通过对cotilting余模的性质和理论的研究,得到关于cotilting余模的一些有意义的结果.     

Gorenstein tilting pairs

Abstract

Let A be an n-Gorenstein ring. Employing the theory developed by Enochs on the existence of Gorenstein preenvelopes and precovers, we introduce the concept of Gorenstein tilting pair. Moreover, we give a simple characterization on Gorenstein tilting pair, which shows that Gorenstein cotilting and tilting modules are special examples of Gorenstein tilting pair.     

Weak Morita Duality

Abstract

The notion of a weak Morita duality between subcategories of Mod-R and S-Mod is introduced and investigated. This concept includes dualities induced by cotilting modules over finite dimensional algebras, the R-dual for both noetherian rings of injective dimension ≤ 1 and the Matlis duality domains, as well as the first author's generalized Morita duality. Moreover it yields a “cotilting theorem” dual to the classical tilting theorem, and a characterization of generalized Morita duality.     

余代数上余倾斜余模的结构

研究了余代数上余倾斜余模的结构特征,证明了每个余倾斜余模都可以写成不可分解的两两非同构的余模的直和形式,每个余倾斜余模包含所有的内射不可分解模作为直和项.最后构造了余倾斜余模的两个例子.