A theorem of Grothendieck states that any smooth reductive algebraic group over a field $k$ admits a maximal torus over $k$. My question concerns what happens for schemes.



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> Let $S$ be a scheme and let $G$ be a smooth reductive group scheme over $S$. Does $G$ admit a maximal torus over $S$?
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Given that this is Grothendieck we are talking about, I imagine if he knew how to prove the result over schemes he would have done so, in particular I'm willing to believe that the answer is "no". However I don't know any explicit counter-examples.


I have a specific application in mind where my scheme $S$ is not too badly behaved, for example I can assume that $S$ is Noetherian, affine, regular with $\mathrm{Pic}(S)=0.$ If anyone knows any positive results in such special cases, I would also be very interested.



Does every reductive group scheme admit a maximal torus?
Counterexamples exist even over $S = \mathrm{Spec} \mathbb{Z}$. See, for instance, Lemma 1.1 and Example 6.2 of [http://math.stanford.edu/~conrad/papers/redgpZ.pdf](http://math.stanford.edu/%7Econrad/papers/redgpZ.pdf) or other places of that paper. If $\mathscr{G}$ in Example 6.2 there had a maximal torus over $\mathbb{Z}$, then that maximal torus would be split, contradicting the anisotropy of $G\_{\mathbb{R}}$.


