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

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