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

We prove that a quadratic A[T]-module Q with Witt index (Q/TQ)⩾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.