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Projective Representations II. Generalized chain geometries

Andrea Blunck, Hans Havlicek

Abstract

In this paper, projective representations of generalized chain geometries are investigated, using the concepts and results of part I. In particular, we study under which conditions such a projective representation maps the chains of a generalized chain geometry $Σ(F,R)$ to reguli; this mainly depends on how the field $F$ is embedded in the ring $R$. Moreover, we determine all bijective morphisms of a certain class of generalized chain geometries with the help of projective representations.

Projective Representations II. Generalized chain geometries

Abstract

In this paper, projective representations of generalized chain geometries are investigated, using the concepts and results of part I. In particular, we study under which conditions such a projective representation maps the chains of a generalized chain geometry to reguli; this mainly depends on how the field is embedded in the ring . Moreover, we determine all bijective morphisms of a certain class of generalized chain geometries with the help of projective representations.

Paper Structure

This paper contains 5 sections, 6 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Transversals
  3. Reguli
  4. Chains
  5. An application

Key Result

Proposition 2.2

For $u\in U\setminus\{0\}$ the following statements are equivalent: In this case, $\alpha:R\to K$ with $u^{\rho_a}=a^\alpha u$ is a homomorphism of rings, and $T$ is a transversal of ${\mathbb P}(R)^\Phi$ exactly if $\alpha$ is surjective, or, equivalently, if $Ku$ is a cyclic submodule of the right $R$-module $U$.

Theorems & Definitions (12)

  • Definition 2.1
  • Proposition 2.2
  • Theorem 3.1
  • Remark 3.2
  • Theorem 3.3
  • Example 4.1
  • Example 4.3
  • Theorem 4.4
  • Example 4.5
  • Theorem 5.1
  • ...and 2 more