Projective Representations I. Projective lines over rings

TL;DR

This paper explores how the projective line over a ring can be represented within a projective space over a field, using bimodules, and examines the geometric properties of these representations.

Contribution

It introduces a new framework for representing projective lines over rings via bimodules in projective spaces, extending classical concepts to non-commutative settings.

Findings

Distant points correspond to complementary subspaces in the representation.

Non-distant points can also have complementary images in certain cases.

The representation generalizes classical projective geometry to rings with 1.

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.