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Extending the Concept of Chain Geometry

Andrea Blunck, Hans Havlicek

Abstract

We introduce the chain geometry $Σ(K,R)$ over a ring $R$ with a distinguished subfield $K$, thus extending the usual concept where $R$ has to be an algebra over $K$. A chain is uniquely determined by three of its points, if, and only if, the multiplicative group of $K$ is normal in the group of units of $R$. This condition is not equivalent to $R$ being a $K$-algebra. The chains through a fixed point fall into compatibility classes which allow to describe the residue at a point in terms of a family of affine spaces with a common set of points.

Extending the Concept of Chain Geometry

Abstract

We introduce the chain geometry over a ring with a distinguished subfield , thus extending the usual concept where has to be an algebra over . A chain is uniquely determined by three of its points, if, and only if, the multiplicative group of is normal in the group of units of . This condition is not equivalent to being a -algebra. The chains through a fixed point fall into compatibility classes which allow to describe the residue at a point in terms of a family of affine spaces with a common set of points.

Paper Structure

This paper contains 4 sections, 13 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Definition and basic results
  3. Compatibility of chains
  4. The residue at a point

Key Result

Lemma 2.1

Let $p,q\in{\mathbb P}(R)$ be different points of $\Sigma(K,R)$. Then $p{\mathrel{{\triangle}}} q$ holds exactly if there is a chain ${\mathcal{D}}\in{\mathfrak C}(K,R)$ joining $p$ and $q$.

Theorems & Definitions (17)

  • Lemma 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Remark 2.6
  • Remark 2.7
  • Example 2.8
  • Proposition 2.9
  • Theorem 3.1
  • Theorem 3.2
  • ...and 7 more