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Nonextended quadratic forms over polynomial rings over power series rings

1981
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Proceedings of the American Mathematical Society
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If R is a complete discrete valuation ring, then every quadratic space over R[T] is extended from R. We here show by an example that a corresponding result for higher-dimensional complete regular local rings is not valid. 1 + Y2T2 = (1 + cT)(l + bT), 1 + (iX +jY)T-kXYT2 = (1 + dT)(l + bT). From the first equation we get c = -b, -b2 = Y2. This implies that b = \Y, A G H[[X, Y]]. From the second equation we get d + b = iX +jY, db = -kXY so that we have (iX +jY)X= -(kX + Y).

doi:10.1090/s0002-9939-1981-0627667-5
fatcat:i2cnvln6yrh4hhozegtyyuz2bi