Homotopy invariance of coherent Witt groups
type: Item
instance of: scholarly article
title: Homotopy invariance of coherent Witt groups
zbMATH Open document ID: 1028.11025
author: Stefan G. Gille
published in: Mathematische Zeitschrift
publication date: 2003-08-17T00:00:00Z
review text: 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} \] {}.
Mathematics Subject Classification ID: 11E81
Mathematics Subject Classification ID: 19G12
zbMATH DE Number: 1965489
zbMATH Keywords: homotopy invariance
zbMATH Keywords: triangular Witt groups
zbMATH Keywords: coherent Witt groups
zbMATH Keywords: Gorenstein rings
reviewed by: Manfred Kolster
MaRDI profile type: Publication
full work available at URL: s00209-003-0489-z
OpenAlex ID: W1551072870
DOI: 10.1007/S00209-003-0489-Z
description: scientific article; zbMATH DE number 1965489