A unified Kummer-Artin-Schreier sequence
type: Item
instance of: scholarly article
title: A unified Kummer-Artin-Schreier sequence
zbMATH Open document ID: 0608.12026
DOI: 10.1007/BF01458325
author: William C. Waterhouse
published in: Mathematische Annalen
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164259
review text: For \(b\) in a commutative ring \(R_0\), let \(G\{b\}\) be the affine group scheme defined by \(\{(x,y)\mid x(1-by)=1\}\). This is smooth with connected fibers of dimension one, and it turns out that \(H^1(R,G\{b\})\) is a relative Picard group familiar from \(K\)-theory for any \(R_0\)-algebra \(R\). Now let \(p\) be a prime, and let \(\zeta\) be a \(p\)-th root of unity. Then over \(R_0 = \mathbb{Z}[\zeta]\) there is an exact sequence \[ 1\longrightarrow \mathbb{Z}/p\mathbb{Z}\longrightarrow G\{1-\zeta \}\longrightarrow G\{(1-\zeta)^p\}\longrightarrow 1. \] The cohomology for this sequence reduces to the Kummer sequence when \(1/p\in R\), and it reduces to the Artin-Schreier sequence when \(p=0\) in \(R\). For \(p=2\) (and arbitrary \(R)\) it yields the description of étale quadratic algebras previously derived by adhoc methods.
Mathematics Subject Classification ID: 12G05
Mathematics Subject Classification ID: 14L15
zbMATH DE Number: 3983328
zbMATH Keywords: affine group scheme
zbMATH Keywords: relative Picard group
zbMATH Keywords: Kummer sequence
zbMATH Keywords: Artin-Schreier sequence
zbMATH Keywords: étale quadratic algebras
MaRDI profile type: Publication
cites work: Quadratic forms over semilocal rings
cites work: Q5562621
cites work: The Picard invariant of a principal homogeneous space
cites work: The Group of Unramified Kummer Extensions of Prime Degree
cites work: Diskriminanten und Picard-Invarianten freier quadratischer Erweiterungen
cites work: Q3871006
cites work: The group of quadratic extensions
cites work: One-dimensional affine group schemes
OpenAlex ID: W1991500730
description: scientific article; zbMATH DE number 3983328