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Projective Representations I. Projective lines over rings

Andrea Blunck, Hans Havlicek

Abstract

We discuss representations of the projective line over a ring $R$ with 1 in a projective space over some (not necessarily commutative) field $K$. Such a representation is based upon a $(K,R)$-bimodule $U$. The points of the projective line over $R$ are represented by certain subspaces of the projective space $P(K,U\times U)$ that are isomorphic to one of their complements. In particular, distant points go over to complementary subspaces, but in certain cases, also non-distant points may have complementary images.

Projective Representations I. Projective lines over rings

Abstract

We discuss representations of the projective line over a ring with 1 in a projective space over some (not necessarily commutative) field . Such a representation is based upon a -bimodule . The points of the projective line over are represented by certain subspaces of the projective space that are isomorphic to one of their complements. In particular, distant points go over to complementary subspaces, but in certain cases, also non-distant points may have complementary images.

Paper Structure

This paper contains 5 sections, 13 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. The projective line over a ring
  3. Homomorphisms
  4. Projective representations
  5. Examples

Key Result

Proposition 2.1

Let $(a,b)\in R^2$ be admissible, and let $s\in R$. Put $(a',b'):=s(a,b)$. Then

Theorems & Definitions (26)

  • Proposition 2.1
  • Proposition 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Proposition 3.1
  • Proposition 3.2
  • Remark 3.3
  • Lemma 3.4
  • ...and 16 more