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?

  • 2
    $\begingroup$ Are you assuming flatness? Otherwise it is easy to find counterexamples. $\endgroup$ Apr 11, 2021 at 11:58
  • 1
    $\begingroup$ 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

1 Answer 1


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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.