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)
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 [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
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