On the Galois and flat cohomology of unipotent algebraic groups over local and global function fields. I
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.
scientific article; zbMATH DE number 5304649
scientific article