On the Galois and flat cohomology of unipotent algebraic groups over local and global function fields. I
type: Item
instance of: scholarly article
title: On the Galois and flat cohomology of unipotent algebraic groups over local and global function fields. I
zbMATH Open document ID: 1149.11020
author: Nguyêñ Quôć Thǎńg
author: Nguyên Duy Tân
published in: Journal of Algebra
publication date: 2008-07-29T00:00:00Z
review text: Let \(k\) be a field and \(G\) be a smooth connected unipotent algebraic group defined over \(k\). If, in addition, \(k\) is perfect, it is a result of \textit{M. Rosenlicht} [Ann. Mat. Pura Appl., IV. Ser. 43, 25--50 (1957; Zbl 0079.25703); ibid. 61, 97--120 (1963; Zbl 0126.16901)] that there is a normal series of \(k\)-subgroups of \(G\) with factors isomorphic over \(k\) to the additive group \(G_a\). As a consequence of this, if \(k\) is perfect, the first Galois cohomology \(H^1(k,G)\) is trivial. This result is false if \(k\) is not perfect. Unipotent groups enjoy certain unusual properties over non-perfect fields. For example, Serre constructed a certain function field \(k\) (of positive characteristic) and a unipotent group \(G\) over \(k\) such that \(H^1(k,G)\) is non-trivial. \textit{J. Oesterlé} in [Invent. Math. 78, 13--88 (1984; Zbl 0542.20024)] developed a theory of unipotent groups over local and global fields and has given an example of a unipotent group \(G\) over a locally compact field of positive characteristic whose first Galois cohomology is infinite. In the paper under review, the authors study unipotent groups over non-perfect fields along the lines of the above mentioned references and investigate certain finiteness properties of Galois and flat cohomology of unipotent groups over non-perfect fields, generalizing some of the existing results of Serre, Raynaud and Oesterlé. They also prove some local-global principle for \(H^1(k,G)\) when \(G\) is unipotent. The paper is well-written and has many results of interest to algebraic group theorists.
Mathematics Subject Classification ID: 11E72
Mathematics Subject Classification ID: 20G10
zbMATH DE Number: 5304649
zbMATH Keywords: Galois cohomology
zbMATH Keywords: flat cohomology
zbMATH Keywords: unipotent groups
zbMATH Keywords: local-global principle
reviewed by: Maneesh Thakur
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2008.01.027
OpenAlex ID: W2004665869
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DOI: 10.1016/J.JALGEBRA.2008.01.027
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description: scientific article; zbMATH DE number 5304649