General Case (gpt-5.5)

Brief Log of Running

• created_at_utc: “2026-05-02T22:43:02.214171+00:00”,
• updated_at_utc: “2026-05-02T22:43:02.214187+00:00”,
• Model: Rethlas (Generator GPT-5.5 xhigh, Verifier GPT-5.4 xhigh)
• statement_type: prove_or_disprove
• statement: Let $G$ be a (not necessarily reductive) smooth connected affine group over a (not necessarily perfect) field $k$. Let $K$ be an algebraic field extension of $k$. Let $\mathbb{X}_ \ast(G):=\mathrm{Hom}_ {k\text{-groups}}(\mathbb{G}_ m,G)$ $k$-homomorphisms. Let $G_ K$ be the base change of $G$ to $K$. Prove (or disprove with a counterexample) that $\mathbb{X}_ \ast(G)/G(k)\rightarrow\mathbb{X}_ \ast(G_ K)/G(K)$ is injective. (You can take for granted that this statement is true if $k$ is a perfect field.)

lemma lem:split_unipotent_h1

statement

Let $k$ be a field and let $U$ be a $k$-split smooth connected unipotent $k$-group. Then $$ H^1(k,U)=1 $$ for nonabelian fppf cohomology. Consequently, if $$ 1\to U\to E\xrightarrow{\pi} Q\to 1 $$ is an exact sequence of smooth affine $k$-groups, then the map $$ E(k)\to Q(k) $$ is surjective.

proof

Since $U$ is $k$-split, it has a composition series by smooth connected normal $k$-subgroups whose successive quotients are $k$-isomorphic to $\mathbf G_ a$. Induction on the length of such a series, using the exact sequence in nonabelian fppf cohomology and $H^1(k,\mathbf G_ a)=0$, gives $H^1(k,U)=1$.

For the consequence, the fiber over any $q\in Q(k)$ is a left $U$-torsor over $\operatorname{Spec} k$. Its cohomology class is trivial, so the fiber has a $k$-point. Hence $E(k)\to Q(k)$ is surjective.

lemma lem:split_unipotent_reduction

statement

Let $G$ be a smooth connected affine $k$-group, let $$ U=R_ {us,k}(G) $$ be its maximal $k$-split smooth connected unipotent normal $k$-subgroup, and put $$ \overline G=G/U. $$ Assume that for every algebraic extension $K/k$ the natural map $$ \mathbb X_ \ast(\overline G)/\overline G(k)\to \mathbb X_ \ast(\overline G_ K)/\overline G(K) $$ is injective. Then for every algebraic extension $K/k$ the natural map $$ \mathbb X_ \ast(G)/G(k)\to \mathbb X_ \ast(G_ K)/G(K) $$ is injective.

proof

Let $\pi:G\to \overline G$ be the quotient map, and let $\lambda,\mu\in \mathbb X_ \ast(G)$ become $G(K)$-conjugate. Then $\overline\lambda=\pi\circ\lambda$ and $\overline\mu=\pi\circ\mu$ become $\overline G(K)$-conjugate. By the assumed injectivity for $\overline G$, there is $\overline g\in \overline G(k)$ such that $$ \overline\mu=\operatorname{Int}(\overline g)\circ\overline\lambda. $$ By Lemma lem:split_unipotent_h1, $G(k)\to\overline G(k)$ is surjective, so choose $g\in G(k)$ lifting $\overline g$. Replacing $\lambda$ by $\operatorname{Int}(g)\circ\lambda$, we reduce to the case $$ \pi\circ\lambda=\pi\circ\mu=:\nu:\mathbf G_ m\to\overline G. $$

Form the pullback $k$-group $$ E:=\mathbf G_ m\times_ {\nu,\overline G,\pi}G. $$ Then $E$ is smooth connected affine and fits into an exact sequence $$ 1\to U\to E\xrightarrow{q}\mathbf G_ m\to 1. $$ The two cocharacters $\lambda$ and $\mu$ are the same thing as two $k$-homomorphic sections $$ s_ \lambda(t)=(t,\lambda(t)),\qquad s_ \mu(t)=(t,\mu(t)) $$ of $q$.

We use the following cited result.

Complete cited statement: Theorem 4.2.9 in Structure and classification of pseudo-reductive groups states: “Any two maximal split $k$-tori in a smooth connected affine $k$-group $G$ are conjugate under $G(k)$.”

paper_id: Structure and classification of pseudo-reductive groups
theorem_id: Theorem 4.2.9
arXiv id:

The images $s_ \lambda(\mathbf G_ m)$ and $s_ \mu(\mathbf G_ m)$ are split one-dimensional $k$-tori in $E$. They are maximal split $k$-tori: a split torus in $E$ maps injectively to $\mathbf G_ m$, because the kernel $U$ is unipotent and contains no nontrivial torus. By the cited theorem, there is $e\in E(k)$ such that $$ e\,s_ \lambda(\mathbf G_ m)\,e^{-1}=s_ \mu(\mathbf G_ m). $$ Let $a=q(e)\in k^\times=\mathbf G_ m(k)$. Since $s_ \mu(a)$ centralizes $s_ \mu(\mathbf G_ m)$, the element $$ u:=s_ \mu(a)^{-1}e $$ lies in $U(k)$ and still carries $s_ \lambda(\mathbf G_ m)$ onto $s_ \mu(\mathbf G_ m)$. Moreover $q\circ\operatorname{Int}(u)\circ s_ \lambda=q\circ s_ \lambda$. The restriction $q|_ {s_ \mu(\mathbf G_ m)}$ is an isomorphism onto $\mathbf G_ m$, so the only section of $q$ with image $s_ \mu(\mathbf G_ m)$ is $s_ \mu$. Thus $$ s_ \mu=\operatorname{Int}(u)\circ s_ \lambda, $$ and hence $\mu=\operatorname{Int}(u)\circ\lambda$. Therefore $\lambda$ and $\mu$ are $G(k)$-conjugate.

lemma lem:geometric_normalizer_conjugacy

statement

Let $F$ be an algebraically closed field, let $H$ be a smooth connected affine $F$-group, and let $T\subset H$ be a maximal torus. If $\lambda,\mu\in X_ \ast(T)$ are $H(F)$-conjugate, then they are conjugate by an element of $N_ H(T)(F)$.

proof

Choose $g\in H(F)$ such that $\mu=\operatorname{Int}(g)\circ\lambda$. Let $$ C=Z_ H(\mu(\mathbf G_ m))^\circ. $$ The standard torus-centralizer theorem says that the centralizer of a torus in a smooth affine group is smooth; hence $C$ is a smooth connected affine $F$-group. Both $T$ and $gTg^{-1}$ contain $\mu(\mathbf G_ m)$, so they are contained in $C$. They are maximal tori of $C$, since any larger torus in $C$ would be a larger torus in $H$.

Over the algebraically closed field $F$, maximal tori are maximal split tori. Applying Theorem 4.2.9 from Structure and classification of pseudo-reductive groups to the smooth connected affine group $C$, choose $c\in C(F)$ with $$ c\,gTg^{-1}c^{-1}=T. $$ Then $n:=cg\in N_ H(T)(F)$, and because $c$ centralizes $\mu(\mathbf G_ m)$, $$ \operatorname{Int}(n)\circ\lambda =\operatorname{Int}(c)\circ\mu =\mu. $$ Thus $\lambda$ and $\mu$ are $N_ H(T)(F)$-conjugate.

lemma lem:wound_relative_dominant_representatives

statement

Let $H$ be a smooth connected affine $k$-group such that $R_ {u,k}(H)$ is $k$-wound. Let $S\subset H$ be a maximal split $k$-torus, and let $P$ be a minimal pseudo-parabolic $k$-subgroup containing $S$. Let $$ {}^k\Phi=\Phi(H/R_ {u,k}(H),S) $$ be the relative root system with positive system ${}^k\Phi^+=\Phi(P/R_ {u,k}(H),S)$, and set $$ C=\{\nu\in X_ \ast(S)\mid \langle a,\nu\rangle\ge 0 \text{ for all }a\in{}^k\Phi^+\}. $$ Then:

1. every element of $X_ \ast(S)$ is $N_ H(S)(k)$-conjugate to a unique element of $C$;
2. if $\lambda,\mu\in X_ \ast(S)$ become $H(K)$-conjugate over an algebraic extension $K/k$, then they have the same unique representative in $C$. In particular, $\lambda$ and $\mu$ are $N_ H(S)(k)$-conjugate.

proof

We use the following cited results.

Complete cited statement: Proposition 5.3.1 in Structure and classification of pseudo-reductive groups states that if $S$ is a maximal split $k$-torus in a smooth connected affine $k$-group $G$, then $$ W(G,S):=N_ G(S)/Z_ G(S) $$ is a constant finite étale $k$-group and the inclusion $$ N_ G(S)(k)/Z_ G(S)(k)\hookrightarrow W(G,S)(k) $$ is an equality.

paper_id: Structure and classification of pseudo-reductive groups
theorem_id: Proposition 5.3.1
arXiv id:

Complete cited statement: Theorem 5.3.2(i) in Structure and classification of pseudo-reductive groups states that for a smooth connected affine $k$-group $G$, a maximal split $k$-torus $S$, and a minimal pseudo-parabolic $k$-subgroup $P$ containing $S$, the set $$ {}^k\Phi=\Phi(G/R_ {u,k}(G),S) $$ is a root system in its $\mathbf Q$-span in $X(S)_ \mathbf Q$, the subset $$ \Phi(P/R_ {u,k}(G),S) $$ is a positive system of roots, and the natural map $$ N_ G(S)(k)/Z_ G(S)(k)\to W({}^k\Phi) $$ is an isomorphism.

paper_id: Structure and classification of pseudo-reductive groups
theorem_id: Theorem 5.3.2(i)
arXiv id:

Complete cited statement: Theorem 5.3.2(iv) in Structure and classification of pseudo-reductive groups states that if $R_ {u,k}(G)$ is $k$-wound, then the root system $\Phi(G,S)$ consists of the nontrivial $S$-weights on $\operatorname{Lie}(G)$.

paper_id: Structure and classification of pseudo-reductive groups
theorem_id: Theorem 5.3.2(iv)
arXiv id:

By Theorem 5.3.2(i), $N_ H(S)(k)/Z_ H(S)(k)$ is identified with the Weyl group of the root system ${}^k\Phi$. The usual root-system argument shows that each Weyl orbit on $X_ \ast(S)$ has a unique representative in the closed dominant chamber $C$. Proposition 5.3.1 realizes these Weyl elements by $k$-points of $N_ H(S)$. This proves part (1).

For part (2), let $\lambda^+,\mu^+\in C$ be the unique representatives of $\lambda$ and $\mu$ from part (1). Since the conjugations from $\lambda$ to $\lambda^+$ and from $\mu$ to $\mu^+$ are already defined over $k$, the cocharacters $\lambda^+$ and $\mu^+$ are still $H(K)$-conjugate. Replacing $\lambda,\mu$ by $\lambda^+,\mu^+$, we may assume from now on that $$ \lambda,\mu\in C. $$

Let $k_ s$ be a separable closure of $k$. Choose a maximal $k_ s$-split torus $$ T\subset H_ {k_ s} $$ containing $S_ {k_ s}$. Choose a minimal pseudo-parabolic $k_ s$-subgroup $$ B\subset P_ {k_ s} $$ containing $T$. Let $$ \Phi_ {\mathrm{abs}}=\Phi(H_ {k_ s}/R_ {u,k_ s}(H_ {k_ s}),T) $$ be the corresponding absolute root system, with positive system $\Phi_ {\mathrm{abs}}^+=\Phi(B/R_ {u,k_ s}(H_ {k_ s}),T)$.

For $\nu\in X_ \ast(S)\subset X_ \ast(T)$ and $\alpha\in\Phi_ {\mathrm{abs}}$, $$ \langle \alpha,\nu\rangle=\langle \alpha|_ S,\nu\rangle. $$ By Theorem 5.3.2(iv), because the $k$-wound property of $R_ {u,k}(H)$ is preserved under separable extension, the relative and absolute roots are exactly the nontrivial weights of $S$ and $T$ on the corresponding Lie algebras. Since $B\subset P_ {k_ s}$, every $\alpha\in\Phi_ {\mathrm{abs}}^+$ restricts either to $0$ or to an element of ${}^k\Phi^+$, and every element of ${}^k\Phi^+$ arises as the nonzero restriction of some element of $\Phi_ {\mathrm{abs}}^+$. Hence, for cocharacters in $X_ \ast(S)$, dominance for ${}^k\Phi^+$ is equivalent to dominance for $\Phi_ {\mathrm{abs}}^+$. Thus $\lambda$ and $\mu$ are also $\Phi_ {\mathrm{abs}}^+$-dominant.

Let $\Omega$ be an algebraic closure containing both $K$ and $k_ s$. Since $\lambda$ and $\mu$ are $H(K)$-conjugate, they are $H(\Omega)$-conjugate. The torus $T_ \Omega$ is a maximal torus of $H_ \Omega$. By Lemma lem:geometric_normalizer_conjugacy, $\lambda$ and $\mu$ are conjugate by an element of $N_ {H_ \Omega}(T_ \Omega)(\Omega)$.

By Proposition 5.3.1 applied over $k_ s$ to the maximal split torus $T$, the finite étale group $N_ {H_ {k_ s}}(T)/Z_ {H_ {k_ s}}(T)$ is constant; therefore purely inseparable extension from $k_ s$ to $\Omega$ adds no new Weyl elements. By Theorem 5.3.2(i), this Weyl group is $W(\Phi_ {\mathrm{abs}})$. Thus $\lambda$ and $\mu$ lie in the same $W(\Phi_ {\mathrm{abs}})$-orbit.

Every Weyl-group orbit in a root system has a unique element in the closed dominant chamber. Since both $\lambda$ and $\mu$ are $\Phi_ {\mathrm{abs}}^+$-dominant, we get $$ \lambda=\mu. $$ Undoing the initial $N_ H(S)(k)$-conjugations to dominant representatives proves that $\lambda$ and $\mu$ have the same unique representative in $C$, and hence are $N_ H(S)(k)$-conjugate.

lemma lem:wound_unipotent_case

statement

Let $H$ be a smooth connected affine $k$-group such that $R_ {u,k}(H)$ is $k$-wound. For every algebraic extension $K/k$, the natural map $$ \mathbb X_ \ast(H)/H(k)\to\mathbb X_ \ast(H_ K)/H(K) $$ is injective.

proof

Let $\lambda,\mu\in\mathbb X_ \ast(H)$ become $H(K)$-conjugate. We use again Theorem 4.2.9 from Structure and classification of pseudo-reductive groups: any two maximal split $k$-tori in a smooth connected affine $k$-group are conjugate under $k$-rational points.

Choose a maximal split $k$-torus $S\subset H$. Since the image of any $k$-cocharacter is a split $k$-torus, Theorem 4.2.9 lets us conjugate $\lambda$ and $\mu$, separately by elements of $H(k)$, so that both lie in $X_ \ast(S)$. These conjugated cocharacters are still $H(K)$-conjugate.

Choose a minimal pseudo-parabolic $k$-subgroup $P$ containing $S$. Applying Lemma lem:wound_relative_dominant_representatives to $H,S,P$, the two cocharacters in $X_ \ast(S)$ are $N_ H(S)(k)$-conjugate. Hence the original $\lambda$ and $\mu$ are $H(k)$-conjugate.

theorem thm:main

statement

Let $G$ be a (not necessarily reductive) smooth connected affine group over a (not necessarily perfect) field $k$. Let $K$ be an algebraic field extension of $k$. Let $\mathbb{X}_ \ast(G):=\mathrm{Hom}_ {k\text{-groups}}(\mathbb{G}_ m,G)$ $k$-homomorphisms. Let $G_ K$ be the base change of $G$ to $K$. Prove (or disprove with a counterexample) that $\mathbb{X}_ \ast(G)/G(k)\rightarrow\mathbb{X}_ \ast(G_ K)/G(K)$ is injective. (You can take for granted that this statement is true if $k$ is a perfect field.)

proof

We prove that the displayed map is injective; hence there is no counterexample.

Let $$ U=R_ {us,k}(G) $$ be the maximal $k$-split smooth connected unipotent normal $k$-subgroup of $G$, and set $\overline G=G/U$.

We use the following cited result.

Complete cited statement: Corollary 3.12 in The structure of solvable groups over fields states that for any smooth connected affine $k$-group $G$, $$ R_ {us,k}(G)=R_ {u,k}(G)_ {\mathrm{split}}. $$ In particular, $$ R_ {u,k}(G)/R_ {us,k}(G) $$ is $k$-wound.

paper_id: The structure of solvable groups over fields
theorem_id: Corollary 3.12
arXiv id:

Applying this result to $G$, the unipotent radical of $\overline G$ is $$ R_ {u,k}(\overline G)=R_ {u,k}(G)/R_ {us,k}(G), $$ which is $k$-wound. Therefore Lemma lem:wound_unipotent_case proves injectivity for $\overline G$.

Finally Lemma lem:split_unipotent_reduction lifts injectivity from $\overline G$ to $G$. Thus, if two $k$-homomorphisms $$ \mathbf G_ m\to G $$ become conjugate by an element of $G(K)$, then they were already conjugate by an element of $G(k)$. Equivalently, $$ \mathbb X_ \ast(G)/G(k)\to \mathbb X_ \ast(G_ K)/G(K) $$ is injective.