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There are a few standard sources for learning about linear algebraic groups, such as Humphreys' book "Linear Algebraic Groups" and Borel's book by the same title. Both are not written in the language of group schemes. Humphreys does everything over an algebraically closed field, whereas Borel already introduces fields of definition and studies rationality properties through the text. This makes Borel's book more comprehensive but unfortunately the older language of algebraic geometry and in particular $k$-structures is a bit disconcerting especially if one is used to and prefers the language of schemes and functor of points.

Hence this reasoning led me to choose Humphreys as a source to learn this material, who only works over an algebraically closed field, which has the advantage that the language difference isn't really an issue.

I'd read through parts of Borel on things that I need, but now I feel like I'd like to have a deeper understanding about rationality, $\overline{k}/k$ forms of groups over some base field, etc. without having to go through Borel since the language is awkward from my point of view. So, I'm looking for references:

An ideal reference would cover the following material, via the language of schemes and group schemes: everything that Borel (or Springer, in his book) does on rationality and fields of definition, and perhaps some more recent results as well, seasoned with plenty of Galois cohomology. Some elegant papers using the group scheme language could also be a substitute.

In other words:

Are there any good references on the basic rationality properties (such as the existence of maximal tori and Borel subgroups defined over the base field) for someone who as already read the basic material covered in a book like Humphreys, that uses the language of schemes?

I don't have too high hopes for a book, but perhaps there is some survey paper I've missed? I find the seeming lack of sources on this material in a modern language very frustrating.

Note that Question 17662 is different in that I'm not looking for a book that covers all of the classification of reductive groups; for this, Humphrey's book is fine together with Conrad's fine notes.

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    $\begingroup$ Have you looked at the notes from Brian Conrad's courses on linear algebraic groups? They are of course thoroughly modern. In particular, there are course notes covering the entirety of the first semester which might have some of what you want. $\endgroup$
    – Keenan Kidwell
    Commented Apr 27, 2013 at 2:18
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    $\begingroup$ They're not yet complete, so I don't know if they will eventually have everything you want, but Milne is in the process of writing a long book on algebraic groups from the perspective of group schemes. My impression is that he wants to do the stuff that Borel does (+ more) from a modern perspective. What's written is here : jmilne.org/math/CourseNotes/ala.html $\endgroup$
    – Andy Putman
    Commented Apr 27, 2013 at 2:49
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    $\begingroup$ What do you mean by "(such as existence of....Borel subgroups defined over the base field)"? There are plenty of examples that have no Borel subgroup over the ground field (i.e., are not quasi-split). Do you mean rational conjugacy of minimal parabolics, that every connected reductive group has a unique quasi-split inner form, or something else? $\endgroup$
    – user30180
    Commented Apr 27, 2013 at 13:37
  • $\begingroup$ @ayanta: yes sorry, I was being sloppy. I am aware that not all groups are split or quasisplit. The result on the existence of split and quasisplit forms is one thing for example that I‘d like to see. @Kidwell: thanks, I will take a look at these notes! $\endgroup$
    – user1437
    Commented Apr 27, 2013 at 16:19
  • $\begingroup$ @Jason: The existence of the split form (usually called the Existence Theorem) is proved in those Luminy notes over $\mathbf{Z}$. The existence and uniqueness of a quasi-split inner form is also treated there, in the cohomological exercises. $\endgroup$
    – user30180
    Commented Apr 27, 2013 at 16:25

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