On the Witt group of the punctured spectrum of a regular semilocal ring

Abstract

Let $R$ be a regular semilocal ring of dimension $4q+1⩾5$ which contains $\frac{1}{2}$, $l⩾1$ the number of maximal ideals of $R$ which are assumed to be all of the same height, and $U$ the punctured spectrum of $R$, that is, $SpecR$ without the maximal ideals. We show that the Witt ring $W\left(U\right)$ of $U$ has $l$ non-trivial generators $E_1,\text{…},E_l$ as $W\left(R\right)$-algebra, which satisfy $E_i{E}_j=0$ for all $1⩽i,j⩽l$. If $R$ is integral, then these generators become trivial over the fraction field $K$ of $R$. In particular, the natural morphism $W\left(U\right)⟶W\left(K\right)$ is not injective.