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.



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> **Question.** Does it follow that $G$ has a maximal $r$-dimensional $A^\times$-torus?
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Maximal torus of linear algebraic group over a ring
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](https://thuses.com/algebraic-geometry/some-examples-of-algebraic-groups/) that gives a description of such $G \to S$ by taking centralizers of certain elements in $\text{SL}\_{2,S}$.


