# Are maximal tori conjugate Zariski-locally?

Let $$S$$ be a scheme and let $$G\to S$$ be a reductive group scheme. Then $$G$$ admits a maximal torus etale-locally, and any two maximal tori are conjugate etale-locally, by Theorem 3.2.6 and Corollary 3.2.7 in [B. Conrad, Reductive group schemes] http://math.stanford.edu/~conrad/papers/luminysga3.pdf.

As pointed out in the book, below Proposition 3.1.9 and at the beginning of Chapter 4, the existence of maximal tori can be achieved Zariski-locally (cf. [SGA3, XIV, 3.20]). My question is the following

Let $$T$$ and $$T'$$ be maximal tori in $$G$$ defined over $$S$$, are they conjugate Zariski-locally?

In my case, $$T$$ is split over $$S$$, and $$S$$ is affine with $$Pic(S)=0$$.

I would appreciate any remarks!

• Take $S=\mathrm{Spec}(\mathbb{R})$ and $G=\mathrm{SL}_2$. Then the standard diagonal torus and $\mathrm{SO}_2$ are maximal tori, and they are not conjugate over $\mathbb{R}$. – Laurent Moret-Bailly Jan 12 at 19:09
• @Laurent Moret-Bailly, thank you so much! It's simple and clear! – Evgeny Jan 12 at 19:39
• What happens if one also assumes that $T$ and $T'$ are Zariski-locally isomorphic? – Uriya First Jan 13 at 8:06
• If both tori are split, the answer seems to be "yes", for any two Borel subgroups are Zariski-locally conjugate, and maximal tori inside Borel subgroups are also conjugate (even globally if the base is affine). – Victor Petrov Jan 13 at 15:39