Book contents
- Frontmatter
- Contents
- Introduction
- Terminology, conventions, and notation
- Part I Constructions, examples, and structure theory
- Part II Standard presentations and their applications
- Part III General classification and applications
- Part IV Appendices
- A Background in linear algebraic groups
- B Tits' work on unipotent groups in nonzero characteristic
- C Rational conjugacy in connected groups
- References
- Index
C - Rational conjugacy in connected groups
from Part IV - Appendices
Published online by Cambridge University Press: 05 July 2011
- Frontmatter
- Contents
- Introduction
- Terminology, conventions, and notation
- Part I Constructions, examples, and structure theory
- Part II Standard presentations and their applications
- Part III General classification and applications
- Part IV Appendices
- A Background in linear algebraic groups
- B Tits' work on unipotent groups in nonzero characteristic
- C Rational conjugacy in connected groups
- References
- Index
Summary
Let G be a smooth connected affine group over a field k. In [BoTi2], Borel and Tits announced (without proof) some remarkable results generalizing important theorems when G is reductive. Among these are the G(k)-conjugacy of maximal k-split k-tori, maximal k-split smooth connected unipotent k-subgroups, and minimal pseudo-parabolic k-subgroups, as well as the Bruhat decomposition for G(k) (relative to a choice of minimal pseudo-parabolic k-subgroup). In this appendix we use §§2.1-3.5 and Appendix B to prove these results, following the ideas outlined in [Ti3, §§2-3] (with some scheme-theoretic improvements). We also give some generalizations in §C.4 for group schemes locally of finite type over a field.
Pseudo-completeness
A key observation is that the coset space G/P modulo a pseudo-parabolic k-subgroup P satisfies the following variant of the valuative criterion for properness.
Definition C.1.1 A scheme X over a field k is pseudo-complete over k if it is finite type and separated and X(R) = X(K) for any discrete valuation ring R over k with fraction field K and residue field separable over k.
For any pseudo-complete X, if C is a smooth curve over k and c ∈ C is a closed point such that k(c)/k is separable then any k-morphism C - {c} → X uniquely extends to a k-morphism C → X.
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- Chapter
- Information
- Pseudo-reductive Groups , pp. 494 - 524Publisher: Cambridge University PressPrint publication year: 2010