Extendability of quadratic modules over a polynomial extension of an equicharacteristic regular local ring

TL;DR

This paper proves that quadratic modules over polynomial extensions of equicharacteristic regular local rings are extended from the base ring if they have sufficiently large Witt index, generalizing previous results and establishing a local-global principle for elementary orthogonal transformations.

Contribution

It extends the known results on quadratic modules by proving their extendability over polynomial extensions in a more general setting and establishes a local-global principle for DSER elementary orthogonal transformations.

Findings

Quadratic modules with large Witt index are extended from the base ring.

Established a Local-Global Principle for DSER elementary orthogonal transformations.

Generalized previous theorems to a broader class of regular local rings.

Abstract

We prove that a quadratic $A[T]$-module $Q$ with Witt index ($Q/TQ$)$ \geq d$, where $d$ is the dimension of the equicharacteristic regular local ring $A$, is extended from $A$. This improves a theorem of the second named author who showed it when $A$ is the local ring at a smooth point of an affine variety over an infinite field. To establish our result, we need to establish a Local-Global Principle (of Quillen) for the Dickson--Siegel--Eichler--Roy (DSER) elementary orthogonal transformations.