The forms of the Witt group schemes
type: Item
instance of: scholarly article
title: The forms of the Witt group schemes
zbMATH Open document ID: 0902.14032
author: Changchun Li
published in: Journal of Algebra
publication date: 1997-02-23T00:00:00Z
review text: Throughout this paper, let \(p\) be a fixed prime number and let \(\mathbb{F}_p\) be the prime field of characteristic \(p\). We assume that all Witt group schemes are with respect to this \(p\) and all rings are algebras over \(\mathbb{F}_p\). The purpose of this paper is to study the forms of Witt group schemes. For any scheme \(S\), we denote by \(W_{n,S}\) \((W_{\infty,S})\) the Witt group scheme of finite length \(n\) (resp. of the length of infinity); for each \(n\in\mathbb{N}\), we indicate the \(n\)-dimensional vector group scheme (resp. affine space) over \(S\) by \(G^n_{\alpha,S}\) (resp. \(\mathbb{A}^n_S)\). If \(S=\text{Spec} B\) for some commutative ring \(B\), then we write \(W_{n,B}\) (resp. \(W_{\infty,B}\), \(G^n_{\alpha,B}\), and \(\mathbb{A}^n_B)\) for \(W_{n,S}\) (resp. \(W_{\infty,S}\), \(G^n_{\alpha,S}\), and \(\mathbb{A}^n_S)\). As we know, the three terms, the one-dimensional vector group scheme, the additive group scheme, and \(W_1\), indicate the same thing. Let \(S\) be a scheme. Suppose that \(G\) and \(G'\) are two group schemes over \(S\). We say that \(G'\) is an \(S\)-form of \(G\) provided that there exists a faithfully flat and quasi-compact morphism \(S'\to S\) such that \(G_{S'} \simeq G_{S'}'\) as \(S'\)-group schemes. Let \(A\) be an integral domain with fraction field \(K\) and \(G\) be an affine flat group scheme over \(A\). We say that \(G\) is a model of \(G_K\). Definition. Let \(A\) be a commutative algebra over \(\mathbb{F}_p\). It is called an \(F\)-ring, provided that \(A\) is integrally closed domain and the injection map \(A\hookrightarrow A^{p^{-1}}\) is flat. If furthermore \(\text{Pic} (A)=0\) we call \(A\) an \(FP\)-ring. For \(FP\)-rings, the basic examples are any field of characteristic \(p\), any polynomial algebra of a finite number of indeterminates over a field of characteristic \(p\), and any local \(F\)-ring. This paper consists of four sections: In section 1, we study the forms of vector group schemes, e…
Mathematics Subject Classification ID: 14L15
Mathematics Subject Classification ID: 14G15
zbMATH DE Number: 954749
zbMATH Keywords: characteristic \(p\)
zbMATH Keywords: forms of Witt group schemes
MaRDI profile type: Publication
full work available at URL: jabr.1996.0369
OpenAlex ID: W2092532754
DOI: 10.1006/JABR.1996.0369
Recommended article: Witt groups of Grassmann varieties
Recommended article: Witt groups of Hermitian forms over a Brauer-Severi variety
Recommended article: Witt groups of Hermitian forms over \(\ast\)-fields
Recommended article: Some canonical pairings of Witt groups of forms
Recommended article: Exact sequences of Witt groups of equivariant forms
Recommended article: The Witt group of real algebraic varieties
Recommended article: Derived Witt groups of a scheme
Recommended article: Q5555033
Recommended article: Q4934312
Recommended article: Witt Groups of Varieties and the Purity Problem
description: scientific article; zbMATH DE number 954749