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
DOI: https://doi.org/10.1090/S1056-3911-06-00422-X
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 Mirkovic 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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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
Email: gprasad@umich.edu

Jiu-Kang Yu
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Email: jyu@math.purdue.edu

Brian Conrad
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Email: bdconrad@umich.edu

DOI: https://doi.org/10.1090/S1056-3911-06-00422-X
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

American Mathematical Society