Abstract

If Γ is a thick and residually connected Dn-geometry, n ≥ 4, it is well known that Γ is denned over a unique ground division ring which is commutative. Here we give an elementary proof of the commutativity based on the construction of null polarities in the projective subspaces of Γ, for n = 4.

Introduction

Geometries over a diagram of type Dn, n ≥ 4 are completely classified and well known. They are often discussed together with polar spaces since they stem essentially from quadrics of maximal Witt index. However, the theory of polar spaces is rather intricate and it requires well over a hundred printed pages as we can see from [11] or [4]. A straightforward theory for Dn geometries is much simpler and shorter. Actually, it may come next to the theory of projective geometries by its simplicity. This was observed quite early ([10]).

We are developing a selfcontained theory of Dn-geometries from the definition to the classification. To the best of our knowledge such a treatment is not yet available. It may be useful in various directions, in particular the preparation of extensions to geometries over slightly more general diagrams.

The main result of the theory is as follows.