# Maximal torus of linear algebraic group over a ring

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?

• Are you assuming flatness? Otherwise it is easy to find counterexamples. Apr 11, 2021 at 11:58
• 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 $X$. Apr 11, 2021 at 12:12

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 $$\text{SL}_{2,S}$$.