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Extensions of étale by connected group spaces


Author: David B. Jaffe
Journal: Trans. Amer. Math. Soc. 335 (1993), 155-173
MSC: Primary 14L15; Secondary 14E20
DOI: https://doi.org/10.1090/S0002-9947-1993-1150015-4
MathSciNet review: 1150015
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Abstract: The main theorem, in rough terms, asserts the following. Let $ K$ and $ D$ be group spaces over a scheme $ S$. Assume that $ K$ has connected fibers and that $ D$ is finite and étale over $ S$ . Assume that there exists a single finite, surjective, étale, Galois morphism $ \overline S \to S$ which decomposes (scheme-theoretically) every extension of $ D$ by $ K$. Let $ \pi = \operatorname{Aut}(\overline S /S)$. Then group space extensions of $ D$ with kernel $ K$ are in bijective correspondence with pairs $ (\xi ,\chi )$ consisting of a $ \pi $-group extension

$\displaystyle \xi :1 \to K(\overline S) \to X \to D(\overline S ) \to 1$

and a $ \pi $-group homomorphism $ \chi :X \to \operatorname{Aut}(\overline K )$ which lifts the conjugation map $ X \to \operatorname{Aut}(K(\overline S ))$ and which agrees with the conjugation map $ K(\bar S) \to \operatorname{Aut}(\overline K )$. In this way, the calculation of group space extensions is reduced to a purely group-theoretic calculation.

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  • [1] M. Artin, The implicit function theorem in algebraic geometry, Algebraic Geometry (Internat. Colloq., Tata Inst. Fund. Res., Bombay, 1968), Oxford Univ. Press, London, 1969, pp. 13–34. MR 0262237
  • [2] M. Artin, Versal deformations and algebraic stacks, Invent. Math. 27 (1974), 165–189. MR 0399094, https://doi.org/10.1007/BF01390174
  • [3] Hyman Bass, Algebraic 𝐾-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0249491
  • [4] J. E. Bertin, Généralités sur les preschémas en groupes, Exposé VI$ _{B}$ in Séminaire de Géométrie Algébrique (SGA 3), Lecture Notes in Math., vol. 151, Springer-Verlag, New York, 1970, pp. 318-410.
  • [5] M. Demazure and P. Gabriel, Groupes algébriques (Tome I), North-Holland, Paris, 1970.
  • [6] A. Grothendieck, Fondements de la géométrie algébrique (extraits du Séminaire Bourbaki 1957-1962), Secrétariat mathématique, Paris, 1962.
  • [7] Revêtements étales et groupe fondamental, Lecture Notes in Mathematics, Vol. 224, Springer-Verlag, Berlin-New York, 1971 (French). Séminaire de Géométrie Algébrique du Bois Marie 1960–1961 (SGA 1); Dirigé par Alexandre Grothendieck. Augmenté de deux exposés de M. Raynaud. MR 0354651
  • [8] A. Grothendieck and J. A. Dieudonné, Eléments de géométrie algébrique II, Inst. Hautes Études Sci. Publ. Math. 8 (1961).
  • [9] -, Eléments de géométrie algébrique IV (part two), Inst. Hautes Études Sci. Publ. Math. 24 (1965).
  • [10] -, Eléments de géométrie algébrique IV (part four), Inst. Hautes Études Sci. Publ. Math. 32 (1967).
  • [11] L. K. Hua and I. Reiner, Automorphisms of the unimodular group, Trans. Amer. Math. Soc. 71 (1951), 331–348. MR 0043847, https://doi.org/10.1090/S0002-9947-1951-0043847-X
  • [12] Donald Knutson, Algebraic spaces, Lecture Notes in Mathematics, Vol. 203, Springer-Verlag, Berlin-New York, 1971. MR 0302647
  • [13] Serge Lang and John Tate, Principal homogeneous spaces over abelian varieties, Amer. J. Math. 80 (1958), 659–684. MR 0106226, https://doi.org/10.2307/2372778
  • [14] Daniel A. Marcus, Number fields, Springer-Verlag, New York-Heidelberg, 1977. Universitext. MR 0457396
  • [15] Hideyuki Matsumura and Masayoshi Miyanishi, On some dualities concerning abelian varieties, Nagoya Math. J. 27 (1966), 447–462. MR 0206010
  • [16] James S. Milne, Étale cohomology, Princeton Mathematical Series, vol. 33, Princeton University Press, Princeton, N.J., 1980. MR 559531
  • [17] J. S. Milne, Abelian varieties, Arithmetic geometry (Storrs, Conn., 1984) Springer, New York, 1986, pp. 103–150. MR 861974
  • [18] F. Oort, Commutative group schemes, Lecture Notes in Mathematics, vol. 15, Springer-Verlag, Berlin-New York, 1966. MR 0213365
  • [19] Michel Raynaud, Faisceaux amples sur les schémas en groupes et les espaces homogènes, Lecture Notes in Mathematics, Vol. 119, Springer-Verlag, Berlin-New York, 1970 (French). MR 0260758
  • [20] -, Groupes algébriques unipotents. Extensions entre groupes unipotents et groupes de type multiplicatif, Exposé XVII in Séminaire de Géométrie Algébrique (SGA 3), Lecture Notes in Math., vol. 152, Springer-Verlag, New York, 1970, pp. 532-631.
  • [21] Maxwell Rosenlicht, Extensions of vector groups by abelian varieties, Amer. J. Math. 80 (1958), 685–714. MR 0099340, https://doi.org/10.2307/2372779
  • [22] Jean-Pierre Serre, Algebraic groups and class fields, Graduate Texts in Mathematics, vol. 117, Springer-Verlag, New York, 1988. Translated from the French. MR 918564

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1993-1150015-4
Keywords: Group schemes
Article copyright: © Copyright 1993 American Mathematical Society