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 Spec(R) by removing the maximal ideals of height d, then the natural map \(\operatorname W(R)\to \operatorname W(U)\) is injective. Secondly, given a regular integral scheme X of finite Krull dimension, consider Q its function field and the natural map \(\operatorname W(X)\to \operatorname W(Q)\). 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.