环的左自由正规化扩张和拟Excellent扩张

本文引进了环的左自由正规化扩张和拟Ｅｘｃｅｌｅｎｔ扩张，并将环的Ｅｘｃｅｌｅｎｔ扩张的一些性质推广到了左自由正规化扩张和拟Ｅｘｃｅｌｅｎｔ扩张，主要讨论了几乎Ｎｏｅｔｈｅｒ性，正则性，凝聚性和半遗传性．     

Gorenstein Modules and Auslander Categories

In this article, some new characterizations of Gorenstein projective, injective, and flat modules over commutative noetherian local rings are given.     

正规化扩张和摸

设环S是环R的正规化扩张.本文讨论了R—模与S—模两者间的若干相关性质.     

Finite Normalizing Extensions of FSN Rings

Fully semiprimary Noetherian bimodules and their bimodule extensions are examined. In the presence of incomparability of the link graph of prime ideals, certain bimodule extensions preserve the fully semiprimary property. In particular, a finite normalizing extension ring of a fully semiprimary Noetherian ring is also fully semiprimary as a bimodule over the base ring. It is shown that the extension ring is itself a fully semiprimary ring. An application to crossed products over finite groups is given.     

Track Extensions of Categories and Cohomology

We introduce the notion of n-fold track extensions of a category C by a natural system D and prove that such extensions represent classes in the cohomology of C with coefficients in D introduced by Baues–Wirsching. This generalizes a result of Huebschmann on the cohomology of groups.     

Scalar Extensions of Triangulated Categories

Given a triangulated category ${{\mathcal T}}$ over a field K and a field extension L/K, we investigate how one can construct a triangulated category ${{\mathcal T}}_L$ over L. Our approach produces the derived category of the base change scheme X _L if ${{\mathcal T}}$ is the bounded derived category of a smooth projective variety over K and the field extension is finite and Galois. We also investigate how the dimension of a triangulated category behaves under scalar extensions.     

Universal Central Extensions in Semi-Abelian Categories

Basing ourselves on Janelidze and Kelly’s general notion of central extension, we study universal central extensions in the context of semi-abelian categories. We consider a new fundamental condition on composition of central extensions and give examples of categories which do, or do not, satisfy this condition.     

相对于半对偶化模的Gorenstein同调维数与Auslander范畴

设$R$是一个局部noether环. 我们在本文中研究了相对于半对偶化模$C$的Gorenstein投射, 内射与平坦模. 给出了$C$-Gorenstein同调维数与$\hat{R}$的Auslander范畴之间的关系.     

Hopf Algebra Extensions of Group Algebras and Tambara-Yamagami Categories

We determine the structure of Hopf algebras that admit an extension of a group algebra by the cyclic group of order 2. We study the corepresentation theory of such Hopf algebras, which provide a generalization, at the Hopf algebra level, of the so called Tambara-Yamagami fusion categories. As a byproduct, we show that every semisimple Hopf algebra of dimension < 36 is necessarily group-theoretical; thus 36 is the smallest possible dimension where a non group-theoretical example occurs.     

Auslander correspondence

We study Auslander correspondence from the viewpoint of higher-dimensional analogue of Auslander-Reiten theory [O. Iyama, Higher dimensional Auslander-Reiten theory on maximal orthogonal subcategories, Adv. Math. 210 (1) (2007) 22-50 (this issue)] on maximal orthogonal subcategories. We give homological characterizations of higher dimensional analogue of Auslander algebras in terms of global dimension, Auslander-type conditions and so on. Especially we give an answer to a question of M. Artin [M. Artin, Maximal orders of global dimension and Krull dimension two, Invent. Math. 84 (1) (1986) 195-222]. They are also closely related to Auslander's representation dimension of Artin algebras [M. Auslander, Representation dimension of Artin algebras, in: Lecture Notes, Queen Mary College, London, 1971] and Van den Bergh's non-commutative crepant resolutions of Gorenstein singularities [M. Van den Bergh, Non-commutative crepant resolutions, in: The Legacy of Niels Henrik Abel, Springer, Berlin, 2004, pp. 749-770].     

Free Monoid in Monoidal Abelian Categories

We give an explicit construction of the free monoid in monoidal abelian categories when the monoidal product does not necessarily preserve coproducts. Then we apply it to several new monoidal categories that appeared recently in the theory of Koszul duality for operads and props. This gives a conceptual explanation of the form of the free operad, free dioperad and free properad.      

Auslander systems

The authors generalize the dynamical system constructed by
J. Auslander in 1959, resulting in perhaps the simplest family of examples of minimal but not strictly ergodic systems. A characterization of unique ergodicity and mean-L-stability is given. The new systems are also shown to have zero topological entropy and fail to be weakly rigid. Some results on the set of idempotents in the enveloping semigroup are also achieved.

     

Free Modules over Cartesian Closed Topological Categories

The construction of free R-modules over a Cartesian closed topological category X is detailed (where R is a ring object in X), and it is shown that the insertion of generators is an embedding. This result extends the well-known construction of free groups, and more generally of free algebras over a Cartesian closed topological category. Mathematics Subject Classifications (2000) 18D15, 18D35, 18A40.     

The Distribution Free Newsboy Problem: Review and Extensions

We present here a new, very compact, proof of the optimality of Scarf's ordering rule for the newsboy problem where only the mean and the variance of the demand are known. We then extend the analysis to the recourse case, where there is a second purchasing opportunity; to the fixed ordering cost case, where a fixed cost is charged for placing an order; to the case of random yields; and to the multi-item case, where multiple items compete for a scarce resource.     

Some Isoperimetric Inequalities for Kernels of Free Extensions

If G is a hyperbolic group (resp. synchronously or asynchronously automatic group) which can be expressed as an extension of a finitely presented group H by a finitely generated free group, then the normal subgroup H satisfies a polynomial isoperimetric inequality (resp. exponential isoperimetric inequality).     

Rank Gradient and p-gradient of Amalgamated Free Products and HNN Extensions

We calculate the rank gradient and p-gradient of free products with amalgamation over an amenable subgroup and HNN extensions with an amenable associated subgroup. The notion of cost is used to compute the rank gradient of amalgamated free products and HNN extensions. For the p-gradient the Kurosh subgroup theorems for amalgamated free products and HNN extensions will be used.     

Semi-dualizing complexes and their Auslander categories

Let be a commutative Noetherian ring. We study -modules, and complexes of such, with excellent duality properties. While their common properties are strong enough to admit a rich theory, we count among them such, potentially, diverse objects as dualizing complexes for on one side, and on the other, the ring itself. In several ways, these two examples constitute the extremes, and their well-understood properties serve as guidelines for our study; however, also the employment, in recent studies of ring homomorphisms, of complexes ``lying between' these extremes is incentive.

     

Gorenstein injective modules and Auslander categories

In this paper, we study Gorenstein injective modules over a local Noetherian ring R. For an R-module M, we show that M is Gorenstein injective if and only if Hom _R (Ȓ,M) belongs to Auslander category B(Ȓ), M is cotorsion and Ext ⁱ _R (E,M) = 0 for all injective R-modules E and all i > 0. Received: 24 August 2006 Revised: 30 October 2006     

Derived equivalences and Cohen-Macaulay Auslander algebras

Let A and B be Artin R-algebras of finite Cohen-Macaulay type. Then we prove that, if A and B are standard derived equivalent, then their Cohen-Macaulay Auslander algebras are also derived equivalent. And we show that Gorenstein projective conjecture is an invariant under standard derived equivalence between Artin R-algebras.