We prove that a quadratic A[T ]-module Q with Witt index (Q/T Q)≥ 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.
For a unital ring, it is an open question whether flatness of simple modules implies all modules are flat and thus the ring is von Neumann regular. The question was raised by Ramamurthi over 40 years ago [15] who called such rings SF-rings (i.e., simple modules are flat). In this note we show that a SF Steinberg algebra of an ample Hausdorff groupoid, graded by an ordered group, has an aperiodic unit space. For graph groupoids this implies that the graphs are acyclic. Combining with the Abrams-Rangaswamy Theorem [2], it follows that SF Leavitt path algebras are regular, answering Ramamurthi's question in positive for the class of Leavitt path algebras.On the occasion of his 80th birthday to Kulumani M. Rangaswamy whose passion for Mathematics is contagious
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