Vanishing and nilpotence of locally trivial symmetric spaces over regular schemes

Abstract

We prove two results about Witt rings W(m) 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 $W(R)\to W(U)$ is injective. Secondly, given a regular integral scheme X of finite Krull dimension, consider Q its function field and the natural map $W(X)\to W(Q)$. 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 \( \Ker\big(\W(X)\to \W(Q)\big) \) is zero. That is, this kernel is nilpotent. We give upper and lower bounds for the exponent N.