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Let $G$ be a linear algebraic group over a $k$-algebra $A$, where $k$ is an algebraically closed field. Consider the structure morphism $G\rightarrow U={\rm Spec}(A)$. Assume that for every $k$-point of $A$, the algebraic group $G_k$ has a maximal $r$-dimensional $k$-torus.

Question. Does it follow that $G$ has a maximal $r$-dimensional $A^\times$-torus?

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    $\begingroup$ Are you assuming flatness? Otherwise it is easy to find counterexamples. $\endgroup$ Apr 11, 2021 at 11:58
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    $\begingroup$ A possible approach for constructing counterexamples with $G$ reductive: Start with a nonsplit Azumaya algebra $A$ of degree $n$ over a simply-connected affine $k$-variety $X=\mathrm{Spec}\,A$ and take $G=\mathrm{GL}_1(A)$. Then $G$ specializes to $\mathrm{GL}_n$ at every closed point of $X$, and $\mathrm{GL}_n$ contains a maximal $n$-dimensional torus. On the other hand, I would expect $n$-dimensional tori in $G$ to come from connected rank-$n$ finite etale subalgebras of $A$ (via $E\mapsto \mathrm{GL}_1(E)$), which cannot exist by the simple-connectivity of $X$. $\endgroup$ Apr 11, 2021 at 12:12

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The following is a counterexample. Let $S = \operatorname{Spec} k[[t]]$ and $G \to S$ an affine group scheme with special fiber $\mathbf{G}_a$ and generic fiber $\mathbf{G}_m$. The special fiber contains a maximal torus (namely the zero-dimensional torus). But now a zero-dimensional torus in $G$ cannot possibly be maximal because over the generic fiber it is strictly contained in $\mathbf{G}_m$.

Remark: Sean Cotner has written up a nice article that gives a description of such $G \to S$ by taking centralizers of certain elements in $\text{SL}_{2,S}$.

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