Pseudo-reductive Groups

by Brian Conrad , Ofer Gabber , Gopal Prasad

• ISBN13: 9780521195607
• ISBN10: 0521195608
• Format: Hardcover
• Copyright: 2010-08-31
• Publisher: Cambridge University Press

Summary

Pseudo-reductive groups arise naturally in the study of general smooth linear algebraic groups over non-perfect fields and have many important applications. This self-contained monograph provides a comprehensive treatment of the theory of pseudo-reductive groups and gives their classification in a usable form. The authors present numerous new results and also give a complete exposition of Tits' structure theory of unipotent groups. They prove the conjugacy results (conjugacy of maximal split tori, minimal pseudo-parabolic subgroups, maximal split unipotent subgroups) announced by Armand Borel and Jacques Tits, and also give the Bruhat decomposition, of general smooth connected algebraic groups. Researchers and graduate students working in any related area, such as algebraic geometry, algebraic group theory, or number theory, will value this book as it develops tools likely to be used in tackling other problems.

Table of Contents

Introduction	p. xi
Terminology, conventions, and notation	p. xix
Constructions, examples, and structure theory	p. 1
Overview of pseudo-reductivity	p. 3
Comparison with the reductive case	p. 3
Elementary properties of pseudo-reductive groups	p. 11
Preparations for the standard construction	p. 16
The standard construction and examples	p. 25
Main result	p. 34
Weil restriction and fields of definition	p. 36
Root groups and root systems	p. 43
Limits associated to 1 -parameter subgroups	p. 43
Pseudo-parabolic subgroups	p. 59
Root groups in pseudo-reductive groups	p. 67
Representability of automorphism functors	p. 78
Basic structure theory	p. 86
Perfect normal subgroups of pseudo-reductive groups	p. 86
Root datum for pseudo-reductive groups	p. 94
Unipotent groups associated to semigroups of roots	p. 99
Bruhat decomposition and Levi subgroups	p. 114
Classification of pseudo-parabolic subgroups	p. 130
Standard presentations and their applications	p. 147
Variation of (G′, k′/k, T′, C)	p. 149
Absolutely simple and simply connected fibers	p. 149
Uniqueness of (G′, k′/k)	p. 154
Ubiquity of the standard construction	p. 162
Main theorem and central extensions	p. 162
Properties of standardness and standard presentations	p. 170
A standardness criterion	p. 179
Classification results	p. 191
The A1-case away from characteristic 2	p. 192
Types A2 and G2 away from characteristic 3	p. 198
General cases away from characteristics 2 and 3	p. 203
General classification and applications	p. 215
The exotic constructions	p. 217
Calculations in characteristics 2 and 3	p. 217
Basic exotic pseudo-reductive groups	p. 228
Algebraic and arithmetic aspects of basic exotic pseudo-reductive groups	p. 240
Preparations for classification in characteristics 2 and 3	p. 255
Further properties of basic exotic pseudo-reductive groups	p. 255
Exceptional and exotic pseudo-reductive groups	p. 260
The absolutely pseudo-simple groups in characteristic 2	p. 279
TypeA1	p. 280
Root groups and birational group laws	p. 290
Construction of absolutely pseudo-simple groups with a non-reduced root system	p. 299
Classification of absolutely pseudo-simple groups with a non-reduced root system	p. 318
General case	p. 342
Factors with non-reduced root system and the generalized standard construction	p. 342
Classification via generalized standard groups	p. 351
Applications	p. 358
Maximal tori in pseudo-reductive groups	p. 358
Pseudo-semisimplicity	p. 364
Unirationality	p. 368
Structure of root groups and pseudo-parabolic subgroups	p. 372
Appendices	p. 389
Background in linear algebraic groups	p. 391
Review of definitions	p. 392
Some results from the general theory	p. 398
Frobenius morphisms and non-affine groups	p. 402
Split reductive groups: Existence, Isomorphism, and Isogeny Theorems	p. 407
Weil restriction generalities	p. 422
Groups without Levi subgroups	p. 441
Lie algebras and Weil restriction	p. 446
Lie algebras and groups of multiplicative type	p. 457
Tits' work on unipotent groups in nonzero characteristic	p. 473
Subgroups of vector groups	p. 473
Wound unipotent groups	p. 479
The cckp-kernel	p. 482
Torus actions on unipotent groups	p. 485
Rational conjugacy in connected groups	p. 494
Pseudo-completeness	p. 494
Conjugacy results in the smooth affine case	p. 504
Split unipotent subgroups of pseudo-reductive groups	p. 511
Beyond the smooth affine case	p. 519
References	p. 525
Index	p. 527
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