from PART IV - Appendices
Published online by Cambridge University Press: 05 June 2015
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 ksubgroups, 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 and 3] (with some scheme-theoretic improvements). We give some generalizations in §C.4 for group schemes locally of finite type over a field. We also develop a theory of k-root systems and associated root groups in smooth connected affine groups over any field k (with results that are most satisfactory in the pseudo-reductive case, eliminating pseudo-split hypotheses from some results in §3.3).
Nothing in this appendix is used in the main text except for Theorem C.2.3 and Theorem C.2.29, which are used in Chapter 9, and the self-contained Lemma C.4.1, which is used in several places.
Pseudo-completeness
We shall prove that the coset space G/P modulo a pseudo-parabolic ksubgroup 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 of 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.
Proposition C.1.2Let X be a scheme over a field k.
To save this book to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Find out more about the Kindle Personal Document Service.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.