# Commentarii Mathematici Helvetici

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**Volume 78, Issue 1, 2003, pp. 101–115**

**DOI: 10.1007/s000140300004**

Published online: 2003-03-31

Vanishing and nilpotence of locally trivial symmetric spaces over regular schemes

Paul Balmer^{[1]}(1) UCLA, Los Angeles, USA

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 $\operatorname W(R)\to \operatorname W(U)$ 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.

*Keywords: *Witt group of schemes, rational kernel, semi-local ring, triangular Witt groups

Balmer Paul: Vanishing and nilpotence of locally trivial symmetric spaces over regular schemes. *Comment. Math. Helv.* 78 (2003), 101-115. doi: 10.1007/s000140300004