Finite Extensions of Topogenities

Let X = Y Z, Y Z = Ø, < be a topogenity on Y, a topology on X. A (<, )-extension is a topogenity < on X such that < ¦Y = <, (<) = . We establish some properties of (<, )-extensions and construct all of them in the case of a finite Z.     

Extensions of Finite Quantum Groups by Finite Groups

We give a necessary and sufficient condition for two Hopf algebras presented as central extensions to be isomorphic, in a suitable setting. We then study the question of isomorphism between the Hopf algebras constructed in [AG] as quantum subgroups of quantum groups at roots of 1. Finally, we apply the first general result to show the existence of infinitely many non-isomorphic Hopf algebras of the same dimension, presented as extensions of finite quantum groups by finite groups. Partially supported by CONICET, ANPCyT, Secyt (UNC) and Ministerio de Ciencia y Tecnología de la Provincia de Córdoba.     

Finite Quasi-Uniform Extensions. I

Finite extensions of quasi-uniformities for prescribed topologies are examined. We present a necessary and sufficient condition for the existence of such an extension. We investigate the set of all compatible extensions and derive that this is a sup-distributive lattice. Then we examine the cardinalities and the lattice theoretic properties of these sets.     

Finite Quasi-uniform Extensions. II

The lattice of finite extensions of quasi-uniformities for prescribed topologies are examined. An example is presented which shows that in the finite and strict case there can occur that there exists no coarsest compatible extension. It is also verified that inf in the lattice of extensions. In the strict case there exists a coarsest compatible extension if and only if there exists a coarsest extension for X {p} for every p Y - X. It is shown that certain special sup-distributive lattices can be represented by the lattice of extensions. For example every finite distributive lattice is isomorphic to the lattice of extensions for a suitable system.     

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.     

相对整体维数有限的扩张

代数的扩张是指两个代数之间保持单位元的同态映射,设f:B→A是代数的扩张,扩张f的相对整体维数是指所有A-模的相对投射维数的上确界.我们给出了扩张的相对整体维数有限的一个充分必要条件,作为应用,还获得了Hochschild的文[Relative homological algebra, Trans. Am. Math. Soc.,1956,82:246-269]中一个结果的简洁证明.     

Coleman Automorphisms of Extensions of Finite Characteristically Simple Groups by Some Finite Groups

Let G be an extension of a finite characteristically simple group by an abelian group or a finite simple group.It is shown that every Coleman automorphism of G is an inner automorphism.Interest in such automorphisms arises from the study of the normalizer problem for integral group rings.     

Milnor K-Groups and Finite Field Extensions

Let E/F be a finite separable field extension and let m denote the integral part of log₂ [E : F]. David Leep recently showed that if char(F) 2, then for n m the nth power of the fundamental ideal in the Witt ring of E satisfies the equality I ⁿ E = I ^n–m F · I ^m E. The aim of this note is to prove the analogous equality for the Milnor K-groups, that is K _n E = K _n–m F · K _m E for n m. In either of these equalities one may not replace m by m – 1, as examples of certain m-quadratic extensions indicate.     

Conjugacy Classes and Characters for Extensions of Finite Groups

Let H be an extension of a finite group Q by a finite group G. Inspired by the results of duality theorems for tale gerbes on orbifolds, the authors describe the number of conjugacy classes of H that map to the same conjugacy class of Q. Furthermore, a generalization of the orthogonality relation between characters of G is proved.     

A Relative Filtration Index and Fibers of Normal Primes in Extensions of Finite Type

For an extension R A of commutative Noetherian rings the behavior of the associated morphism of topological spaces Spec A Spec R is often measured by its behavior on each of its fibers. Specifically, one studies the 'splitting' (or 'branching') and the 'ramification' that occurs in each fiber. In the classical constructions of faithfully flat analytic extensions (e.g., completion or Henselization) of excellent local rings the splitting and ramification properties are fairly well understood; see EGA IV [6, 18.10], Nagata [13, Sect. 37] or Raynaud [15, Ch. IX]. The strongest results are usually achieved for fibers over a 'normal point' of Spec R, that is, over p Spec R such that R/p is a normal domain [e.g., the property of a normal prime p in a local ring to be 'unibranched', i.e., the Henselization of R/p is a (normal) domain].     

A Characterization of Finite Generalized Dedekind Groups

A generalized Dedekind group is a group G with \({\langle H \leqq G \, | \, H \ntrianglelefteq G \rangle < G}\) . We give a characterization of finite generalized Dedekind groups.     

有限维代数扩张的同构与维数群

对于有限维C＊-代数A，证明了其本质扩张的同构与酉等价是一致的，由此证明了扩张群Ext（A）中的等价类是区分该类扩张代数的完全不变量，并利用Bratteli图计算出它们的维数群．     

On Preservation of Stability for Finite Extensions of Abelian Groups

We characterize preservation of superstability and ω-stability for finite extensions of abelian groups and reduce the general case to the case of p-groups. In particular we study finite extensions of divisible abelian groups. We prove that superstable abelian-by-finite groups have only finitely many conjugacy classes of Sylow p-subgroups. Mathematics Subject Classification: 03C60, 20C05.     

Anisotropic Forms of Higher Degree Under Finite Dimensional Field Extensions

Let F be a field of characteristic 0 or greater than d. We investigate anisotropic forms of degree d > 2 and their behaviour under field extensions of degree coprime to d. A form of degree d over F which permits composition or Jordan composition remains anisotropic under any field extension of degree coprime to d. Other classes of forms are constructed which stay anisotropic under such extensions.     

A Characterization of Nilpotent Injectors in Finite Groups

In this article a class of subgroups of a finite group G, called Q-injectors, is introduced. If G is soluble, the Q-injectors are precisely the injectors of the Fitting sets. A characterization of nilpotent Q-injectors is given as well as a sufficient condition for the solubility of a finite group G, in terms of Q-injectors, which generalizes a well known result.     

More Points Than Expected on Curves over Finite Field Extensions

On average, there are q^r+o(q^r/2) F_q^r-rational points on curves of genus g defined over F_q^r. This is also true if we restrict our average to genus g curves defined over F_q, provided r is odd or r>2g. However, if r=2,4,6,… or 2g then the average is q^r+q^r/2+o(q^r/2). We give a number of proofs of the existence of these q^r/2 extra points, and in some cases give a precise formula, but we are unable to provide a satisfactory explanation for this phenomenon.