Homotopy invariance of coherent Witt groups
The author proves the following homotopy invariance theorem for the Witt group of a regular and separated scheme \(X\) of finite Krull dimension: For every locally free \(\mathcal O_X\)-module \(\mathcal E\) of finite rank with corresponding vector bundle \(\pi: \mathbb V(\mathcal E) \rightarrow X\) the natural homomorphism \[ \pi^*: W(X) \rightarrow W(\mathbb V(\mathcal E)) \] is an isomorphism. The proof of this result uses \textit{P. Balmer}'s triangular Witt theory [\(K\)-Theory 19, 311-363 (2000; Zbl 0953.18003)] to reduce to the affine case as well as the dévissage result from the author's paper [Math. Ann. 322, 103-137 (2002; Zbl 1010.19003)]. As an application the author computes the Witt group of the affine hyperbolic sphere \[ \Sigma_R^{2n-1}=\text{Spec}(R[X_1, \cdots,X_n,T_1,\cdots,T_n]/(1-\sum_{i=1}^n X_iT_i)) \] over a regular local ring \(R\): \[ W(\Sigma_R^{2n-1}) \cong \begin{cases} \quad\quad W(R) \quad\quad\quad\text{if \(n \not\equiv 1 \mod 4\)}\\ W(R) \oplus W(R) \quad \text{if \(n \equiv 1 \mod 4\)}.\end{cases} \] {}.
scientific article; zbMATH DE number 1965489