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I am struggling to understand some of the basic definitions from Borel's book on linear algebraic groups.

Let $ K/k $ be a field extension, Borel defines a $k$-structure of a $K$-scheme as follows: enter image description here

Question 1: Is this equivalent to saying that $X = X' \times_k \operatorname{spec}K$ for some $k$-scheme $X'$?

Next, he defines the rational points:

enter image description here

While it is first defined for $K$-algebras, it seems like we are mostly interested in subfields of $K$, for which I am not sure I understand the definition. After reading another MSE post, and hoping that my understanding of the previous definition is correct,

Question 2: Is it true that $V(k')$ is defined to be $V'(k')$, where $V = V' \times_k \operatorname{spec}K$? And is $V(K)$ just the set underlying $V$?

And finally, Question 3: If $Z\subset V$ is closed (or open) in the $k$-topology, does $Z\subset V(k)$? Why?

MCL
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  • What is a k-topology here? – Hanno Apr 05 '23 at 06:01
  • @Hanno I think it is just the name for a topology that satisfies the conditions of the first definition – MCL Apr 05 '23 at 12:19
  • I apologize for the non-answer, especially as it is probably precisely what you don't want to hear, but I would suggest not investing too much time in trying to decode the classical language here. Start reading Milne's recent book instead referring back to Borel when necessary (e.g. for the more sophisiticated later discussion of non-split groups). In particular, the answer to your Question 1 is yes for all intents and purposes (I didn't actually think through it, but it is what you should take it to mean). – Alex Youcis Apr 06 '23 at 01:19
  • @AlexYoucis Thanks for the advice, I will check out that book – MCL Apr 06 '23 at 15:24

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