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In his book Linear Algebraic Groups, Tonny Springer defines a Borel subgroup of a linear algebraic group $G$ over an algebraically closed field to be a "closed, connected, solvable, subgroup of $G$, which is maximal for these properties".

Apart from having too many commas, this definition always bugged me for the following reason: if we take the closure of a connected solvable subgroup $H \subseteq G$, don't we get a connected solvable subgroup $\overline{H}$ with $H \subseteq \overline{H}$? If so, we wouldn't need to include the word "closed" in the definition of Borel subgroup: a maximal connected solvable subgroup would automatically be closed.

Now that I see the same definition on GroupProps, I feel I must be making a mistake. If so, what is it?

And while I'm at it: is there a generally accepted definition of 'Borel subgroup' for affine group schemes over a field that's not necessarily algebraically closed?

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    $\begingroup$ For your first comment: yes, 'closed' is redundant; indeed, Borel does not include it in his definition (§11.1; p. 147). For your second, perhaps the obvious (and a correct!) definition is a subgroup that becomes Borel after base change to an algebraic closure; is that unsatisfactory? $\endgroup$
    – LSpice
    Oct 26, 2016 at 17:59
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    $\begingroup$ For the utmost generality, Tome XXII, Exemple 5.2.3(a) of the revised SGA3 defines a Borel subgroup of a linear algebraic group over a general scheme fibrewise. $\endgroup$
    – LSpice
    Oct 26, 2016 at 18:14
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    $\begingroup$ @John: The notion goes back to Borel's 1950s work (though of course he didn't at first call it a "Borel subgroup"). After he lectured at Columbia in 1968, Bass wrote up his lecture notes (W.A. Benjamin, 1969). In 1975 my Springer GTM 21 with the same title gave a somewhat more polished version. Following Borel, I defined "Borel subgroup" in 21.3 and also noted that "closed" is redundant. Springer's book came later, still with the same title, followed by Borel's expanded GTM edition of the earlier notes. As L Spice comments, SGA3 extended it all to schemes. $\endgroup$
    – Jim Humphreys
    Oct 26, 2016 at 19:15
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    $\begingroup$ Apropos of @JimHumphreys's comment on the history of the name, and specifically Borel's contribution to it, I have always enjoyed (and maybe even have quoted here before?) the following footnote from Borel and Tits's numdam.org/numdam-bin/fitem?id=PMIHES_1965__27__55_0 , p. 65 / 669, to the definition of a Borel subgroup: "L'un des auteurs insistant pour que l'on adopte cette terminologie, aujourd'hui généralement admise, l'autre auteur s'y résigne." $\endgroup$
    – LSpice
    Oct 26, 2016 at 19:21
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    $\begingroup$ @L Spice: Yes, that's a fine example of what I assume to be Tits' humor, leaving it up to the reader to sort out which author did the insisting. $\endgroup$
    – Jim Humphreys
    Oct 26, 2016 at 19:27

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It's an algebraic subgroup of $G$, not an abstract subgroup of the points $G(k)$ of $G$. In the old days, when algebraic groups were smooth over algebraically closed fields and $G$ was identified with its points $G(k)$, authors used "algebraic subgroup" and "closed subgroup" interchangeably. What Springer really means is that a Borel subgroup is a (smooth) connected solvable algebraic subgroup of $G$ that is maximal for these properties.

For a smooth algebraic group scheme over an arbitrary field $k$, a Borel subgroup is usually defined to be an algebraic subgroup that becomes Borel over the algebraic closure of $k$. Alternatively, one can say that it is a smooth connected solvable algebraic subgroup $B$ such that $G/B$ is complete. This definition works well with respect to change of base field, and has the advantage of avoiding mentioning algebraic closures.

To put it another way, the "closed" in Springer's definition isn't redundant because he was thinking only of closed subgroups.

[I agree with YCor that the footnote was obviously written by Borel, and I find it insulting to Borel to suggest otherwise.]

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    $\begingroup$ As far as I can tell, Springer started with the same hybrid language used by Borel for the treatment of affine/linear algebraic groups over an algebraically closed field $k$. He did try out some modifications in later chapters where the groups are defined over smaller fields, but this is still not the language of SGA3. (On the other hand, Demazure and others using rigorous scheme language never made it all the way through the detailed theory including the Bruhat decomposition and Chevalley classification. Conrad-Gabber-Prasad do more structure theory.) $\endgroup$
    – Jim Humphreys
    Oct 27, 2016 at 19:36
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    $\begingroup$ I'll take this opportunity to piggyback on @JimHumphreys's references by pushing, as I always do since Brian pointed it out to me, Milne's wonderful notes on linear algebraic groups, which use a modern language but are less intimidating than CGP (and, of course, much less than SGA3). $\endgroup$
    – LSpice
    Oct 29, 2016 at 15:05

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