Group scheme with an isotrivial maximal torus

Question

Let $G$ be a reductive group scheme over a normal ring $A$. Then, we know that Zariski locally it admits a maximal torus. Let us assume that it admits a maximal torus after a finite surjective (resp. finite flat) cover, is it possible to replace it by a finite étale cover?

Answer 1

Edit. I realized after the original post that there are even easier examples. These examples also show that the Quillen–Suslin Theorem fails already for smooth affine quadric hypersurfaces (in some sense, the "simplest" smooth affine varieties after affine spaces, where Quillen–Suslin does hold).

Let $\overline{B}$ be a Grassmannian parameterizing $2$-dimensional quotient vector spaces of a fixed vector space $V$ of dimension $n\geq 4$, $$\overline{B}=\operatorname{Grass}_k(V,2),$$ let the rank $2$, locally free $\mathcal{O}_{\overline{B}}$-module be the tautological quotient, $$V\otimes_k \mathcal{O}_{\overline{B}} \twoheadrightarrow \mathcal{E},$$ and let $D\subset \overline{B}$ be a general hyperplane section for the Plücker embedding. For the open complement $B=\overline{B}\setminus D$, all of the arguments in the original post still apply. Moreover, by van Kampen's theorem, the (tame) fundamental group of $B$ is trivial. So that is definitely a better example than in the original post. For completeness, I include below the original post.

Original post. Sorry for the delay. I had an overly elaborate example starting with an Enriques surface in characteristic $2$. In fact, sufficiently general K3 surfaces in characteristic $0$ give even simpler examples.

Let $\overline{B}$ be a K3 surface over $\mathbb{C}$. Recall by Mumford's Theorem that the group $\operatorname{CH}_0(\overline{B})$ is uncountably generated, in fact, it cannot even be parameterized by any (finite dimensional) algebraic variety, in an appropriate and precise sense.

MR0249428
Mumford, D.
Rational equivalence of 0-cycles on surfaces.
J. Math. Kyoto Univ. 9 (1968), 195–204.
https://projecteuclid.org/journals/journal-of-mathematics-of-kyoto-university/volume-9/issue-2/Rational-equivalence-of-0-cycles-on-surfaces/10.1215/kjm/1250523940.full

Moreover, in every ample divisor class on $\overline{B}$, there exists a divisor $D$ whose irreducible components have normalization isomorphic to $\mathbb{P}^1$, cf. Beauville's exposition of the Yau–Zaslow theorem.

MR1682284
Beauville, Arnaud
Counting rational curves on K3 surfaces.
Duke Math. J. 97 (1999), no. 1, 99–108.
https://arxiv.org/abs/alg-geom/9701019

In fact, by a theorem of Beauville–Voisin, the subgroup of $\operatorname{CH}_0(\overline{B})$ generated by all pushforwards of all $0$-cycles on (possibly singular) rational curves in $\overline{B}$ is a cyclic subgroup commensurate with the cyclic subgroup generated by the cycle class $c_2(T_{\overline{B}})$ and containing all cup products of first Chern classes of divisors.

MR2047674
Beauville, Arnaud; Voisin, Claire
On the Chow ring of a K3 surface.
J. Algebraic Geom. 13 (2004), no. 3, 417–426.
https://arxiv.org/abs/math/0111146

Now assume that $\overline{B}$ is a very general K3 surface, so that the Picard group is generated by an ample $\mathcal{O}_{\overline{\mathcal{B}}}$-module $\mathcal{L}$. For definiteness, assume that the degree of $c_1(\mathcal{L})\cap c_1(\mathcal{L})$ equals $2$. Let $D$ be an irreducible, nodal curve in the linear system of $\mathcal{L}$ such that the normalization of $D$ is a rational curve, i.e., an irreducible, nodal curve of arithmetic genus $2$ and geometric genus $0$. For the affine surface $B=\overline{B}\setminus D$, there exists a closed point $p\in B$ such that the cycle class of $\{p\}$ in $\operatorname{CH}_0(\overline{B})$ is not contained in the cyclic subgroup coming from pushforwards of $0$-cycles from $D$. In particular, the cycle class of $\{p\}$ in $\operatorname{CH}_0(B)$ does not equal a cup product of two divisor classes.

Apply Serre's construction to the ideal sheaf $\mathcal{I}_p\subset \mathcal{O}_{\overline{B}}$ with determinant invertible sheaf $\mathcal{L}=\mathcal{O}_{\overline{B}}(D)$, i.e., consider the local-to-global spectral sequence computing $\operatorname{Ext}^*_{\mathcal{O}_{\overline{B}}}(\mathcal{I}_p,\mathcal{L})$. The long exact sequence of low degree terms is $$ 0 \to H^1(\overline{B},\mathcal{L}) \to \operatorname{Ext}^1_{\mathcal{O}_{\overline{B}}}(\mathcal{I}_p,\mathcal{L}) \to L\rvert_p \to H^2(\overline{B},\mathcal{L}) .$$ Here $L\rvert_p$ is the $1$-dimensional vector space that is, essentially, the fiber at $p$ of $\mathcal{L}$. Since the dualizing sheaf on $\overline{B}$ is isomorphic to the structure sheaf, Kodaira vanishing gives that $H^q(\overline{B},\mathcal{L})$ is the zero vector space for $q=1,2$. Thus, there is a unique isomorphism class of a nontrivial extension $\mathcal{E}$ of $\mathcal{I}_p$ by $\mathcal{L}$.

The sheaf $\mathcal{E}$ is a locally free $\mathcal{O}_{\overline{B}}$-module of rank $2$ whose first Chern class equals $c_1(\mathcal{L})$ and whose second Chern class equals the cycle class of $\{p\}$. In particular, the restriction $\mathcal{E}\rvert_B$ is a locally free $\mathcal{O}_B$-module of rank $2$ whose first Chern class is zero and whose second Chern class is the nonzero class of $\{p\}$.

Consider the group scheme $G$ over $\overline{B}$ parameterizing $\mathcal{O}_B$-automorphisms of $\mathcal{E}$ such that the induced automorphism of $\det(\mathcal{E}) \cong \mathcal{L}$ is the identity automorphism. This is an inner form of $\mathbf{SL}_2$ over $\overline{B}$. Since $\mathcal{E}$ is locally free, this inner form is Zariski locally isomorphic to $\mathbf{SL}_2$, and thus has split maximal tori Zariski locally. However, the restriction over $B$ does not have a maximal torus. If it did, the corresponding representation $\mathcal{E}\rvert_B$ would split as a direct sum of locally free sheaves of rank $1$. Then, by the Whitney sum formula, the second Chern class of $\mathcal{E}\rvert_B$ would be in the image of the cyclic subgroup of Beauville–Voisin, i.e., the second Chern class would be a torsion class in $\operatorname{CH}_0(B)$, contrary to the choice of $p$.

Now consider the projective space bundle over $\overline{B}$ that parameterizes invertible quotients of the pullback of $\mathcal{E}$. This is a projective scheme. By Bertini theorems, an intersection of this projective scheme with a sufficiently general hyperplane is a smooth closed subscheme $\overline{A}$ whose projection to $\overline{B}$ is finite surjective, and thus finite flat. In particular, the restriction $A$ over $B$ is a finite, flat cover of $B$. The pullback of $\mathcal{E}\rvert_B$ to $A$ has an invertible quotient. Since $A$ is affine, thus surjection automatically splits, so that the pullback of $\mathcal{E}\rvert_B$ to $A$ is isomorphic to a direct sum of two invertible sheaves. Thus, the pullback of $G$ to $A$ does have a maximal torus.