Horrocks theorem for odd orthogonal groups

TL;DR

This paper extends Horrocks' theorem to the odd elementary orthogonal group, showing that matrices over polynomial rings can be decomposed into simpler orthogonal and elementary matrices when 2 is invertible in the base ring.

Contribution

It proves Horrocks' theorem for the odd orthogonal group, providing a decomposition result over polynomial rings in the context of invertible 2 in the base ring.

Findings

Decomposition of orthogonal matrices over polynomial rings

Extension of Horrocks' theorem to odd orthogonal groups

Applicable when 2 is invertible in the base ring

Abstract

We prove Horrocks' theorem for the odd elementary orthogonal group, which gives a decomposition of an orthogonal matrix with entries from a polynomial ring $R[X]$, over a commutative ring $R$ in which 2 is invertible, as a product of an orthogonal matrix with entries in $R$ and an elementary orthogonal matrix with entries from $R[X]$.