×

On the Galois and flat cohomology of unipotent algebraic groups over non-perfect fields. (English) Zbl 1122.11021

The main object of the note under review is a connected unipotent group \(G\) defined over a local or global field \(k\) of characteristic \(p>0\). The authors are interested in finiteness and certain local-global properties of the Galois and flat cohomology of \(G\). One should note here the striking contrast to the case of zero characteristic, where the first Galois cohomology of \(G\) is trivial. The results announced in the present note provide sufficient conditions for the first flat cohomology of \(G\) to be infinite and local-to-global criteria for the first Galois cohomology to be trivial. The note contains no proofs.

MSC:

11E72 Galois cohomology of linear algebraic groups
20G10 Cohomology theory for linear algebraic groups
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] A. Borel, Linear algebraic groups , 2nd ed., Springer, New York, 1991. · Zbl 0726.20030
[2] S. Bosch, W. Lütkebohmert and M. Raynaud, Néron models , Springer, Berlin, 1990.
[3] M. Demazure and P. Gabriel, Groupes algébriques , tome I, Masson & Cie, Éditeur, Paris, 1970. · Zbl 0203.23401
[4] T. Kambayashi, M. Miyanishi and M. Takeuchi, Unipotent algebraic groups , Lecture Notes in Math., 414, Springer, Berlin, 1974. · Zbl 0294.14022
[5] J. Oesterlé, Nombres de Tamagawa et groupes unipotents en caractéristique \(p\), Invent. Math. 78 (1984), no. 1, 13-88. · Zbl 0542.20024 · doi:10.1007/BF01388714
[6] M. Raynaud, Groupes algébriques unipotents, Extensions entre groupes unipotents et groupes de type multiplicatif, Exp. XVII, in Schémas en groupes , SGA 3 (2), Lecture Notes in Math., vol. 152, Springer, New York, 1970, pp. 532-631.
[7] M. Rosenlicht, Some rationality questions on algebraic groups, Ann. Mat. Pura Appl. (4) 43 (1957), 25-50. · Zbl 0079.25703 · doi:10.1007/BF02411903
[8] M. Rosenlicht, Questions of rationality for solvable algebraic groups over nonperfect fields, Ann. Mat. Pura Appl. (4) 61 (1963), 97-120. · Zbl 0126.16901 · doi:10.1007/BF02412850
[9] P. Russell, Forms of the affine line and its additive group, Pacific J. Math. 32 (1970), 527-539. · Zbl 0199.24502 · doi:10.2140/pjm.1970.32.527
[10] J.-P. Serre, Cohomologie galoisienne , 5th ed., Springer, Berlin, 1994.
[11] N. Q. Thǎńg and N. D. Tan, On the surjectivity of localization maps for Galois cohomology of unipotent algebraic groups over fields, Comm. Algebra 32 (2004), no. 8, 3169-3177. · Zbl 1121.11036 · doi:10.1081/AGB-120039284
[12] J. Tits, Lectures on algebraic groups, Yale Univ., New Haven, 1967.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.