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Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



On quasi-reductive group schemes

Authors: Gopal Prasad and Jiu-Kang Yu; with an appendix by Brian Conrad
Journal: J. Algebraic Geom. 15 (2006), 507-549
Published electronically: March 8, 2006
MathSciNet review: 2219847
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Abstract | References | Additional Information

Abstract: This paper was motivated by a question of Vilonen, and the main results have been used by Mirković and Vilonen to give a geometric interpretation of the dual group (as a Chevalley group over $\mathbb {Z})$ of a reductive group. We define a quasi-reductive group over a discrete valuation ring $R$ to be an affine flat group scheme over $R$ such that (i) the fibers are of finite type and of the same dimension; (ii) the generic fiber is smooth and connected, and (iii) the identity component of the reduced special fiber is a reductive group. We show that such a group scheme is of finite type over $R$, the generic fiber is a reductive group, the special fiber is connected, and the group scheme is smooth over $R$ in most cases, for example when the residue characteristic is not 2, or when the generic fiber and reduced special fiber are of the same type as reductive groups. We also obtain results about group schemes over a Dedekind scheme or a Noetherian scheme. We show that in residue characteristic 2 there are non-smooth quasi-reductive group schemes with generic fiber $\operatorname {SO}_{2n+1}$ and they can be classified when $R$ is strictly Henselian.

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  • Sivaramakrishna Anantharaman, Schémas en groupes, espaces homogènes et espaces algébriques sur une base de dimension 1, Sur les groupes algébriques, Soc. Math. France, Paris, 1973, pp. 5–79. Bull. Soc. Math. France, Mém. 33 (French). MR 0335524, DOI
  • M. Artin, Lipman’s proof of resolution of singularities for surfaces, Arithmetic geometry (Storrs, Conn., 1984) Springer, New York, 1986, pp. 267–287. MR 861980
  • M. Artin and G. Winters, Degenerate fibres and stable reduction of curves, Topology 10 (1971), 373–383. MR 476756, DOI
  • Siegfried Bosch, Werner Lütkebohmert, and Michel Raynaud, Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 21, Springer-Verlag, Berlin, 1990. MR 1045822
  • Armand Borel and Jacques Tits, Homomorphismes “abstraits” de groupes algébriques simples, Ann. of Math. (2) 97 (1973), 499–571 (French). MR 316587, DOI
  • N. Bourbaki, Éléments de mathématique. Fasc. XXXVIII: Groupes et algèbres de Lie. Chapitre VII: Sous-algèbres de Cartan, éléments réguliers. Chapitre VIII: Algèbres de Lie semi-simples déployées, Actualités Scientifiques et Industrielles, No. 1364. Hermann, Paris, 1975 (French). MR 0453824
  • F. Bruhat and J. Tits, Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d’une donnée radicielle valuée, Inst. Hautes Études Sci. Publ. Math. 60 (1984), 197–376 (French). MR 756316
  • T. Chinburg, Minimal models for curves over Dedekind rings, Arithmetic geometry (Storrs, Conn., 1984) Springer, New York, 1986, pp. 309–326. MR 861982
  • P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 75–109. MR 262240
  • [EGA]ega A. Grothendieck: Eléments de géométrie algébrique, Publ. Math. IHES, 4, 8, 11, 17, 20, 24, 28, 32.
  • Schémas en groupes. I: Propriétés générales des schémas en groupes, Lecture Notes in Mathematics, Vol. 151, Springer-Verlag, Berlin-New York, 1970 (French). Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3); Dirigé par M. Demazure et A. Grothendieck. MR 0274458
  • Nathan Jacobson, Basic algebra. I, W. H. Freeman and Co., San Francisco, Calif., 1974. MR 0356989
  • A. J. de Jong, Smoothness, semi-stability and alterations, Inst. Hautes Études Sci. Publ. Math. 83 (1996), 51–93. MR 1423020
  • Hideyuki Matsumura, Commutative algebra, W. A. Benjamin, Inc., New York, 1970. MR 0266911
  • Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR 879273
  • James S. Milne, Étale cohomology, Princeton Mathematical Series, No. 33, Princeton University Press, Princeton, N.J., 1980. MR 559531
  • [MV]MV I. Mirković and K. Vilonen: Geometric Langlands duality and representations of algebraic groups over commutative rings, preprint (2004). [R]R M. Raynaud: Passage au quotient par une relation d’équivalence plate, Proceedings of a Conference on Local Fields, 78–85, Springer-Verlag (1967).
  • T. A. Springer, Linear algebraic groups, 2nd ed., Progress in Mathematics, vol. 9, Birkhäuser Boston, Inc., Boston, MA, 1998. MR 1642713
  • J. Tits, Reductive groups over local fields, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 29–69. MR 546588
  • Adrian Vasiu, Integral canonical models of Shimura varieties of preabelian type, Asian J. Math. 3 (1999), no. 2, 401–518. MR 1796512, DOI
  • William C. Waterhouse, Introduction to affine group schemes, Graduate Texts in Mathematics, vol. 66, Springer-Verlag, New York-Berlin, 1979. MR 547117
  • [Yu]Yu J.-K. Yu: Smooth models associated to concave functions in Bruhat-Tits theory, preprint (2003).

Additional Information

Gopal Prasad
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Address at time of publication: School of Mathematics, Institute for Advanced Study, Einstein Drive, Princeton, New Jersey 08540

Jiu-Kang Yu
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907

Brian Conrad
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
MR Author ID: 637175

Received by editor(s): January 14, 2005
Received by editor(s) in revised form: June 16, 2005
Published electronically: March 8, 2006
Additional Notes: The first author was partially supported by NSF-grant DMS-0100429. The second author was partially supported by NSF-grant DMS-0100678, a Sloan fellowship, and the IHES. The third author was partially supported by NSF-grant DMS-0093542 and a Sloan fellowship.
Dedicated: Dedicated to Pierre Deligne