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I am reading Maxwell Rosenlicht's 1956 paper, "Some Basic Theorems on Algebraic Groups" (American Journal of Mathematics 78(2):404-443) [JSTOR link], and having trouble with some of the notation and language. In particular, since Rosenlicht admits arbitrary ground fields but is writing before Grothendieck's development of scheme theory, I'm confused by what he means by a point of an algebraic variety.

Rosenlicht is working with an algebraic group $G$ defined as "a union of a finite number of disjoint algebraic varieties [I presume "variety" means irreducible in this context] ... together with a group structure [defined by rational maps] ... a field $k$ is called a field of definition of $G$ if it is a field of definition for each component of $G$ and for all of the above rational maps..." (p. 402).

Question: In this context, what are the elements of $G$?

In particular if I want to think of $G$ as a separated scheme of finite type over $k$, is Rosenlicht talking about the closed points? Only the closed points with residue field $k$? Or the closed points of the base change to $\overline k$? Or what?

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  • $\begingroup$ This question may be more suited for MathOverflow. $\endgroup$
    – Servaes
    Apr 2, 2019 at 13:25
  • $\begingroup$ I think he means $\overline{k}$-points, but you might try to work this out in a simple example. Let $G = \mathbb{G}_m/k = \operatorname{Spec} k[x, 1/x]$ be the multiplicative group scheme. Which of the possible definitions makes sense in that case? $\endgroup$
    – Ashvin Swaminathan
    Apr 2, 2019 at 14:44
  • $\begingroup$ It probably means, as Ashvin said, $\overline{k}$-points. If you have particular theorems to ask about perhaps we can help prove them/provide references in modern language. $\endgroup$
    – Alex Youcis
    Apr 3, 2019 at 8:26
  • $\begingroup$ @AshvinSwaminathan - Certainly at least $k$-points and $\overline k$-points make sense. $\endgroup$
    – Ben Blum-Smith
    Apr 3, 2019 at 13:15
  • $\begingroup$ @AlexYoucis - I'm particularly interested in Theorem 2, p. 407, which has come to be known as "Rosenlicht's Theorem". That said, my goal is to be able to read Rosenlicht's own statement and proof, to understand what he was doing. $\endgroup$
    – Ben Blum-Smith
    Apr 3, 2019 at 13:40

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I crossposted this question on MO, per Servaes' comment. It was answered by this user. Rosenlicht is using the Weil foundations for algebraic geometry, which were supplanted by Grothendieck's scheme theory in the 60's. A "point" in this context is a point in an affine patch, with coordinates in a universal domain, which is an algebraically closed extension of $k$ of infinite transcendence degree. (The last link is to an article by Raynaud in the Notices which does a good job explaining what's going on.)

If I am thinking about this correctly, the points can be thought of in scheme-theoretic terms as the closed points of the base change to the universal domain.

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