×

Extensions of finite group schemes, and Hopf Galois theory over a complete discrete valuation ring. (English) Zbl 0737.11038

L. N. Childs has determined all Kummer extensions \(L/K\) of degree \(p\) with \(K\) a \(p\)-adic field such that the corresponding rings of integers give a Hopf Galois extension, i.e. \(\text{Spec}({\mathcal O}_L)\) is a principal homogeneous space over \(\text{Spec}({\mathcal O}_K)\) under a finite \({\mathcal O}_K\)-group. Here we extend his results in several ways: first to cyclic extensions \(L/K\) of degree \(p\) (i.e. \(\zeta_p\in K\) is no longer required), second to arbitrary (non-normal) extensions of degree \(p\), and lastly to cyclic extensions of order \(p^2\). In the second step, it is also shown that all \({\mathcal O}_K\)-groups of order \(p\) do occur as Hopf Galois groups. In the third step, some information about certain \({\mathcal O}_K\)-groups of order \(p^2\) is needed. This amounts to the calculation of some Ext groups, which is done in Part I of the paper. The extensions thus obtained are surprisingly complicated, and their principal homogeneous spaces are discussed. It is no longer true for order \(p^ 2\) that all \({\mathcal O}_K\)-groups occur as Hopf Galois groups. Another difference is the following: For \([L:K]=p\), it suffices to know the ramification number in order to decide whether \({\mathcal O}_L\) is Hopf Galois, but for \([L:K]=p^2\) the knowledge of the two ramification numbers does not suffice for this.

MSC:

11S15 Ramification and extension theory
11S20 Galois theory
16T05 Hopf algebras and their applications
14L30 Group actions on varieties or schemes (quotients)
PDF BibTeX XML Cite
Full Text: DOI EuDML

References:

[1] Cassou-Noguès, P., Taylor, M.: Elliptic functions and rings of integers. (Prog. Math., vol. 66) Basel: Birkhäuser 1986 · Zbl 0621.12012
[2] Chase, S.U., Sweedler, M.: Hopf algebras and Galois theory. (Lect. Notes Math., vol. 97) Berlin Heidelberg New York: Springer 1969 · Zbl 0197.01403
[3] Childs, L.: Taming wild extensions with Hopf algebras. Trans. Am. Math. Soc.304, 111–140 (1987). · Zbl 0632.12013
[4] Childs, L.: On the Hopf Galois theory of separable extensions. Commun. Algebra17, 809–825 (1989) · Zbl 0692.12007
[5] Demazure, M., Gabriel, P.: Groupes algèbriques. Paris: Masson 1970 · Zbl 0203.23401
[6] Edwards, H.: Galois theory. Berlin Heidelberg New York: Springer 1984 · Zbl 0532.12001
[7] Hurley, S.: Galois objects with normal basis for free Hopf algebras of prime degree. J. Algebra109, 292–318 (1987) · Zbl 0622.16005
[8] Larson, R.G.: Orders in Hopf algebras. J. Algebra22, 201–210 (1972) · Zbl 0246.16008
[9] MacLane, S.: Homology. Berlin Heidelberg New York: Springer 1963 · Zbl 0133.26502
[10] Maus, E.: Arithmetisch disjunkte Körper. J. Reine Angew. Math.226, 184–203 (1967) · Zbl 0149.29403
[11] Milne, J.S.: Etale cohomology. Princeton: Princeton University Press 1980 · Zbl 0433.14012
[12] Milne, J.S.: Arithmetic duality theorems. Boston: Academic Press 1986 · Zbl 0613.14019
[13] Raynaud, M.: Schémas en groupes de type (p, ..., p). Bull. Soc. Math. Fr.102, 241–280 (1974) · Zbl 0325.14020
[14] Roberts, L.: The flat cohomology of group schemes of orderp. Am. J. Math.95, 688–702 (1973) · Zbl 0281.14020
[15] Schneider, H.-J.: Endliche algebraische Gruppen. Habilitationsschrift, Universität München (1973)
[16] Schneider, H.-J.: Zerlegbare Erweiterungen affiner Gruppen. J. Algebra66, 569–593 (1979) · Zbl 0452.20040
[17] Serre, J.-P.: Corps locaux. Paris: Hermann 1962 · Zbl 0137.02601
[18] Shatz, S.S.: Principal homogeneous spaces for finite group schemes. Proc. Am. Math. Soc.22, 678–680 (1968) · Zbl 0186.54701
[19] Tate, J.:p-divisible groups. In: Proceedings of a Conf. on Local Fields, pp. 158–183. Driebergen Berlin Heidelberg New York: Springer 1967 · Zbl 0157.27601
[20] Tate, J., Oort, F.: Group schemes of prime order. Ann. Sci. Ec. Norm. Supér.3, 1–21 (1970) · Zbl 0195.50801
[21] Waterhouse, W.: Principal homogeneous spaces and group scheme extensions. Trans. Am. Math. Soc.153, 181–188 (1971) · Zbl 0208.48401
[22] Waterhouse, W.: A unified Kummer-Artin-Schreier sequence. Math. Z.87, 447–451 (1987) · Zbl 0608.12026
[23] Wenninger, C.-H.: Galois-Algebren zu Hopf-Algebren und verallgemeinerte Quaternionen. Dissertation, Universität München (1988)
[24] Wenninger, C.-H.: Corestriction of Galois algebras. J. Algebra144, 359–370 (1991) · Zbl 0737.16010
[25] Wyman, B.F.: Wildly ramified Gamma extensions. Am. J. Math.91, 135–152 (1969) · Zbl 0188.11003
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.