Paul Balmer and Stefan Gille

Let $X$ be a regular separated scheme of finite Krull dimension and let $U^{n}_{X} \subset A^{n}_{X}$ be the punctured affine $n$-space over $X$. We show that the total graded Witt ring of $U^{n}_{X}$ is a free graded module over the total graded Witt ring of $X$ with two generators $1$ and $\epsilon$. The second generator satisfies the equation $\epsilon^{2} = 1$ when $n = 1$ and $\epsilon^{2} = 0$ when $n \geq 2$.

2000 Mathematics Subject Classification: 11E81 19G12.

Keywords: regular scheme, punctured affine space, Witt group.

E-mail:
balmer@math.ethz.ch
gille@math.ethz.ch