Vanishing and nilpotence of locally trivial symmetric spaces over regular schemes

We prove two results about Witt rings W(−) of regular schemes. First, given a semi-local regular ring R of Krull dimension d, if U is the punctured spectrum, obtained from Spec(R) by removing the maximal ideals of height d, then the natural map is injective. Secondly, given a regular integral scheme X of finite Krull dimension, consider Q its function field and the natural map . We prove that there is an integer N, depending only on the Krull dimension of X, such that the product of any choice of N elements in is zero. That is, this kernel is nilpotent. We give upper and lower bounds for the exponent N. Received: December 4, 2001