If $G$ is an algebraic group over a field $F$ (no assumptions on $F$), then is every torus in $G$ defined over $F$ contained in a maximal torus defined over $F$?

  • 12
    $\begingroup$ It seems what you really intend to ask (for smooth affine $F$-groups $G$, no reductivity hypotheses) is if every $F$-torus in $G$ is contained in an $F$-torus that remains maximal after any ground field extension. That is: if an $F$-torus $T \subset G$ is maximal as an $F$-torus then for every extension $F'/F$ is $T_{F'}$ maximal in $G_{F'}$? The answer is "yes", and this is an absolutely fundamental theorem of Grothendieck. By renaming $Z_G(T)^0/T$ as $G$, the real content is to show $G$ admits some $F$-torus remaining maximal over $\overline{F}$; that is what Grothendieck proved. $\endgroup$
    – nfdc23
    Dec 17, 2016 at 22:46
  • 9
    $\begingroup$ It is precisely because of Grothendieck's theorem that the phrase "maximal torus" does not create tremendous confusion when one is making ground field extensions all over the place. It is proved in any textbook on linear algebraic groups that works over general fields; e.g., 18.2(i) in Borel's textbook "Linear Algebraic Groups". $\endgroup$
    – nfdc23
    Dec 17, 2016 at 22:49
  • $\begingroup$ Ah, I had a nontrivial question in mind, asked a trivial question, but you successfully untrivialized it! I didn't think the fact that a maximal F-torus remains maximal after any ground field extension was so nontrivial. I think I got that impression because of the way it was stated in whatever paper/book I was reading. $\endgroup$ Dec 18, 2016 at 0:45


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