Order and topology in projective Hjelmslev planes

The concept of an ordered projective Hjelmslev plane was intuitively introduced by Hjelmslev in Einleitung in die allgemeine Kongruenglehre ([9], [10]).This paper is concerned with formalizing and examing preorderings and orderings for projective Hjelmslev planes. In addition we show that orderings generated topologies of the point and line sets which render the plane a topological Hjelmslev plane ([19], [13]). These planes — unlike the ordinary ordered planes ([18]) — are, due to the existence of infinitesimals, non-archimedian, non-compact and disconnected with the neighbour classes as certain quasi-components.The authors gratefully acknowledge the support of the Natural Sciences and Engineering Research Council of Canada.     

The ternary rings of Desarguesian and Pappian planes — A simple proof using perspectivities

The aim of this paper is to give a direct, simple proof of the well-known theorems — that the Hall ternary ring (R, T) of a Pappian projective plane is a linear ternary ring over a field, and that of a Desarguesian plane is a linear one over a skew field — by making repeated application of the perspectivity theorem in a Pappian plane and the characterization of Desarguesian planes in terms of perspectivities.This is a revised version of the paper The Ternary Ring of a Pappian Plane — A Simple Proof presented at the 49th Conference of the Indian Mathematical Society, held at Madras, December 27–29, 1983.     

Projective Modules and Complete Intersections

Let A be a Noetherian ring of dimension n and P be a projective A module of rank n having trivial determinant. It is proved that if n is even and the image of a generic element g P^* is a complete intersection, then [P] = [Q A] in K₀(A) for some projective A module Q of rank n – 1. Further, it is proved that if n is odd, A is Cohen–Macaulay and [P] = [Q A] in K₀(A) for some projective A module Q of rank n – 1, then P has a unimodular element.     

Maximal subgroups of symmetric groups defined on projective spaces over finite fields

Let PL(n, q) be a complete projective group of semilinear transformations of the projective space P(n–1, q) of projective degree n–l over a finite field of q elements; we consider the group in its natural 2-transitive representation as a subgroup of the symmetric group S(P*(n–1, q)) on the setp*(n–1),q=p(n–1,q)/{O}. In the present note we show that for arbitrary n satisfying the inequality n>4[(qⁿ–1)/(q^n–1–1)] [in particular, for n>4(q +l)] and for an arbitrary substitutiong s (p*(n–1,q))pL(n,q) the group PL(n,q), g contains the alternating group A(P* (n–1,q)). Forq=2, 3 this result is extended to all n3.Translated from Matematicheskie Zametki, Vol. 16, No. 1, pp. 91–100, July, 1974.The author expresses his sincere thanks to M. M. Glukhov for his interest in his work.     

Resolutions over Koszul algebras

In this paper we show that if is a Koszul algebra with Λ₀ isomorphic to a product of copies of a field, then the minimal projective resolution of Λ₀ as a right Λ-module provides all the information necessary to construct both a minimal projective resolution of Λ₀ as a left Λ-module and a minimal projective resolution of Λ as a right module over the enveloping algebra of Λ. The main tool for this is showing that there is a comultiplicative structure on a minimal projective resolution of Λ₀ as a right Λ-module.Received: 14 September 2004     

Weighted Completion of Galois Groups and Galois Actions on the Fundamental Group of P1 - {0,1, ∞}

Fix a prime number . We prove a conjecture stated by Ihara, which he attributes to Deligne, about the action of the absolute Galois group on the pro- completion of the fundamental group of the thrice punctured projective line. Similar techniques are also used to prove part of a conjecture of Goncharov, also about the action of the absolute Galois group on the fundamental group of the thrice punctured projective line. The main technical tool is the weighted completion of a profinite group with respect to a reductive representation (and other appropriate data).     

σ — Inzidenzgruppen

Non-linear incidence groups are considered. The classical model of those is the 3-dimensional double elliptic space over the reals with group ^*/₊, derived from the field of the quaternions. The geometric structure of those incidence groups is that of projective G-fibre spaces [4]. The representation of desarguesian projective incidence groups by nearfields [5] will be extended to those incidence groups.     

Schauder decompositions and the Fremlin projective tensor product of Banach lattices

In this paper, we first introduce a lattice decomposition and finite-dimensional lattice decomposition (FDLD) for Banach lattices. Then we show that for a Banach lattice with FDLD, the following are equivalent: (i) it has the Radon-Nikodym property; (ii) it is a KB-space; (iii) it is a Levi space; and (iv) it is a σ-Levi space. We then give a sequential representation of the Fremlin projective tensor product of an atomic Banach lattice with a Banach lattice. Using this sequential representation, we show that if one of the Banach lattices X and Y is atomic, then the Fremlin projective tensor product has the Radon-Nikodym property (or, respectively, is a KB-space) if and only if both X and Y have the Radon-Nikodym property (or, respectively, are KB-spaces).     

Equivariant Riemann–Roch for G ‐Quasi‐Projective Varieties‐I

Let G denote a complex linear algebraic group. In this paper we establish several forms of equivariant Riemann–Roch valid for the category of Gquasiprojective complex varieties. The main application is to the operation of convolution that appears in the construction of modules over convolution algebras, for example the Hecke algebras associated to a complex reductive group as well as in the representation theory of quantum groups. The paper concludes with a discussion of some of these applications. The results in this paper hold mostly at the level of Grothendieck groups. The second part of this paper will discuss the extension to higher Ktheory in detail.     

Lie symmetry of the Landau-Lifshitz-Gilbert equation and exact linearization in the Minkowski space

In this paper we present a linear representation of the Landau-Lifshitz-Gilbert equation for describing the magnetization of ferromagnetic materials. According to Lies theory, we prove that this equation admits a superposition principle and its formula is derived. The underlying vector space of the Landau-Lifshitz-Gilbert equation is found to be a projective Minkowski space denoted by of which the projective proper orthochronous Lorentz group PSO ^o(3,1) left acts. By the Lie symmetry a group preserving scheme is developed, which improves the computational accuracy and efficiency.     

Modular representations of profinite groups

Our aim is to transfer several foundational results from the modular representation theory of finite groups to the wider context of profinite groups. We are thus interested in profinite modules over the completed group algebra of a profinite group G, where k is a finite field of characteristic p.We define the concept of relative projectivity for a profinite -module. We prove a characterization of finitely generated relatively projective modules analogous to the finite case with additions of interest to the profinite theory. We introduce vertices and sources for indecomposable finitely generated -modules and show that the expected conjugacy properties hold—for sources this requires additional assumptions. Finally we prove a direct analogue of Green’s Indecomposability Theorem for finitely generated modules over a virtually pro-p group.     

Spreads of right quadratic skew field extensions

LetL/K be a right quadratic (skew) field extension and let be a 3-dimensional projective space overK which is embedded in a 3-dimensional projective space overL. Moreover, let be a line of which carries no point of. The main result is that — even whenL orK is a skew field — the following holds true: A Desarguesian spread of is given by the set of all lines of which are indicated by the points of . A spread of arises in this way if, and only if, there exists an isomorphism ofL onto the kernel of the spread such thatK is elementwise invariant. Furthermore, a geometric characterization of right quadratic extensions with a left degree other than 2 and of quadratic Galois extensions is given.Dedicated to O. Giering on the occasion of this 60th birthdayThe term field is to mean a not necessarily commutative field.     

Artin-Schelter regular algebras and categories

Motivated by constructions in the representation theory of finite dimensional algebras we generalize the notion of Artin-Schelter regular algebras of dimension n to algebras and categories to include Auslander algebras and a graded analogue for infinite representation type. A generalized Artin-Schelter regular algebra or a category of dimension n is shown to have common properties with the classical Artin-Schelter regular algebras. In particular, when they admit a duality, then they satisfy Serre duality formulas and the -category of nice sets of simple objects of maximal projective dimension n is a finite length Frobenius category.     

Numerical equivalence defined on Chow groups of Noetherian local rings

In the present paper, we define a notion of numerical equivalence on Chow groups or Grothendieck groups of Noetherian local rings, which is an analogue of that on smooth projective varieties. Under a mild condition, it is proved that the Chow group modulo numerical equivalence is a finite dimensional -vector space, as in the case of smooth projective varieties. Numerical equivalence on local rings is deeply related to that on smooth projective varieties. For example, if Grothendiecks standard conjectures are true, then a vanishing of Chow group (of local rings) modulo numerical equivalence can be proven. Using the theory of numerical equivalence, the notion of numerically Roberts rings is defined. It is proved that a Cohen–Macaulay local ring of positive characteristic is a numerically Roberts ring if and only if the Hilbert–Kunz multiplicity of a maximal primary ideal of finite projective dimension is always equal to its colength. Numerically Roberts rings satisfy the vanishing property of intersection multiplicities. We shall prove another special case of the vanishing of intersection multiplicities using a vanishing of localized Chern characters.     

Zur Darstellung der Automorphismengruppe einer kinematischen Kettengeometrie im Quadrikenmodell

There is a unique projective representation of the group of automorphisms of a geometry () [1] over a kinematic [3] algebra which is compatible with the quadric model [2].     

Factor representations of the infinite spin-symmetric group

Let S() be the group of finite permutations of the sequence of natural numbers. The infinite spin-symmetric group T() is its central Z₂-extension. With the aid of this extension the projective representations of the group S() can be linearized. The paper describes the factor representation of type II₁ of the group T().Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 181, pp. 132–145, 1990.     

Die windschiefen Flächen mit einer stetigen Schar ebener Schattengrenzen II

In part I of this subject it has been shown that each ruled surface of the projective 3-spaceII with a continuous set of plane shade lines (ES-Regelfächen) is a ruled surface ofBlank with two conjugate families of such lines.—In this paper ES-Regelfächen will be constructed by using the specific projective motion of a plane, defined by each continuous set of plane shade lines on a ruled surface (central motion of a plane inII). We show that each such central motion of a shade-plane is the restriction of a one-parametric continuous group of projective collineations of the 3-space to a plane (theorem 5). Using this it is possible to characterize ES-Regelflächen as special surfaces with two conjugate families of plane shade lines (theorems 6 and 7). Finally moulding ruled surfaces in projective, affine, euclidian and non-euclidian 3-spaces are interpreted as ES-Regelflächen, and all those surfaces are listed completely.

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Teil I zu diesem Thema ist in Mh. Math.91, 39–71 (1981) erschienen. Die Numerierung der Abschnitte, Sätze und Fußnoten von Teil II schließt an Teil I an. Die Bezeichnung der auftretenden geometrischen Objekte stimmt mit jener in Teil I überein.     

Zur Theorie Linearer Abbildungen I

The paper is concerned with a uniform geometric definition of linear mappings in a projective or grassmannian space into a projective space. We discuss sufficient conditions for the existence of a linear mapping in a finite dimensional pappian projective space which continues two given linear mappings in complementary subspaces.The subspace spanned by the image set of a linear mapping in the grassmannian of d-dimensional subspaces of an n-dimensional projective space has at most dimension –1.     

On the geometry of the Calogero-Moser system

We discuss a special eigenstate of the quantized periodic Calogero—Moser system associated to a root system. This state has the property that its eigenfunctions, when regarded as multivalued functions on the space of regular conjugacy classes in the corresponding semisimple complex Lie group, transform under monodromy according to the complex reflection representation of the affine Hecke algebra. We show that this endows the space of conjugacy classes in question with a projective structure. For a certain parameter range this projective structure underlies a complex hyperbolic structure. If in addition a Schwarz type of integrality condition is satisfied, then it even has the structure of a ball quotient minus a Heegner divisor. For example, the case of the root system E₈ with the triflection monodromy representation describes a special eigenstate for the system of 12 unordered points on the projective line under a particular constraint.