Some cases of extensions of twisted group schemes over a discrete valuation ring
type: Item
instance of: scholarly article
title: Some cases of extensions of twisted group schemes over a discrete valuation ring
zbMATH Open document ID: 1050.14037
author: Toshiaki Ohno
published in: Journal of Algebra
publication date: 2000-01-01T00:00:00Z
review text: Let \(A\) be a discrete valuation ring with residue field \(k\), and let \(B\) be an unramified quadratic extension of \(A\). Let \(G\) be the nontrivial \(B/A\) -form of the multiplicative group scheme \(\mathbb{G}_{m}.\) Then \(G=\text{ Spec}A[\xi ,\tau]/(n\xi ^{2}-m\xi \tau +\tau ^{2}+m^{2}-4n),\) where \(B=A[\theta]\) and \(\theta ^{2}-m\theta +n=0.\) If we let \(\mathcal{G}\) be the model of \(G\), then \(\mathcal{G}=\text{ Spec}A[v,w]/(\lambda (nv^{2}-mvw+w^{2})-(m^{2}-4n)v)\) for some nonzero \(\lambda \in A.\) The object of this paper is to compute some specific extension groups for \( \mathcal{G}\) and \(G\) using the fppf topology. The most concrete results are as follows: \[ H^{1}(\mathcal{G},\mathbb{G}_{m,A}) \cong \mathbb{Z}/2\mathbb{Z}, \text{ Ext}^{1}(\mathcal{G},\mathbb{G}_{m,A}) \cong \mathbb{Z}/2\mathbb{Z} \] \[ H^{1}(\mathcal{G}^{(\lambda)},G) \cong A^{\times}/\text{Norm}(B^{\times}) \times \mathbb{Z}/2\mathbb{Z}, \text{ Ext}^{1}(\mathcal{G}^{(\lambda)},G) \cong \mathbb{Z}/2\mathbb{Z} \] \[ H^{1}(\mathcal{G},G) \cong A^{\times}/\text{Norm}(B^{\times}), \text{ Ext}^{1}(\mathcal{G},G) =0 \] \[ H^{1}(\mathbb{G}_{a,A},G) \cong A^{\times}/\text{Norm}(B^{\times}) \] where \(\mathbb{G}_{a,A}\) is the additive group scheme over \(A,\) \(( \,) ^{\times}\) represents the group of units, and Norm is the norm map \(B^{\times }\to A^{\times}.\) There is little here to motivate the desire to compute such groups, except the remark that Sekiguchi and Suwa have done similar calculations (such as Ext\(^{1}(\mathcal{G}^{(\lambda)}, \mathcal{G}^{(\mu)})\)). The proofs of these facts require extensive use of cohomological techniques which would be difficult to summarize in this review. In addition to the facts above, the authors obtain the following three results: The model of the \(\prod_{B/A}\mathbb{G}_{m}\) which is the Néron blowup of \(\mathbb{G}_{m,k}\) is the nontrivial extension of \(\mathcal{G}\) by \(\mathbb{G}_{m}.\) The model of the \(\prod_{B/A}\mathbb{G}…
reviewed by: Alan Koch
Mathematics Subject Classification ID: 14L15
zbMATH DE Number: 1471028
zbMATH Keywords: extensions
zbMATH Keywords: twisted group schemes
zbMATH Keywords: multiplicative group scheme
zbMATH Keywords: fppf topology
MaRDI profile type: Publication
full work available at URL: jabr.1999.8086
OpenAlex ID: W1994764277
cites work: Q5602064
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DOI: 10.1006/JABR.1999.8086
description: scientific article; zbMATH DE number 1471028