Vanishing and nilpotence of locally trivial symmetric spaces over regular schemes

Abstract

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 $\mathrm{Spec}(R)$ 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 $\mathrm{Ker}(W(X)\to W(Q))$ is zero. That is, this kernel is nilpotent. We give upper and lower bounds for the exponent $N$.