# Vanishing and nilpotence of locally trivial symmetric spaces over regular schemes

### Paul Balmer

UCLA, Los Angeles, USA

## 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 $W(R)→W(U)$ is injective. Secondly, given a regular integral scheme $X$ of finite Krull dimension, consider $Q$ its function field and the natural map $W(X)→W(Q)$. 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 $Ker(W(X)→W(Q))$ is zero. That is, this kernel is nilpotent. We give upper and lower bounds for the exponent $N$.

## Cite this article

Paul Balmer, Vanishing and nilpotence of locally trivial symmetric spaces over regular schemes. Comment. Math. Helv. 78 (2003), no. 1, pp. 101–115

DOI 10.1007/S000140300004