Lie-Kolchin: smooth connected solvable affine groups over algebraically closed fields are trigonalizable

Every simple rational representation is finite-dimensional by [F6]. I show by induction on $\dim G$ that $\dim V=1$. If $G$ is commutative, [F1] gives $\dim V=1$. Otherwise $G\mathrm{\ne }1$ and [F2] provides the smooth connected closed normal subgroup $N=DG$ with $\dim N<\dim G$, solvable as a subgroup of the solvable group $G$; by the induction hypothesis applied to $N$, the group $N$ is trigonalizable.

The restricted $N$-module $V$ need not be simple; choose a least-dimensional nonzero $N$-submodule as in [F3]. Since $N$ is trigonalizable it is a character line, so there is a character $\chi$ of $N$ with $V_{\chi }\mathrm{\ne }0$; for $g\in G(k)$ and $n\in N(k)$ one computes $n\cdot (g\cdot v)=g\cdot (g^{-1}ng\cdot v)=\chi (g^{-1}ng)g\cdot v$, so $g\cdot V_{\chi }=V_{{\chi }^g}$ for the character ${\chi }^g(n)=\chi (g^{-1}ng)$. Thus $G(k)$ permutes the finite set $S$ of characters $\chi$ of $N$ with $V_{\chi }\mathrm{\ne }0$.

Fix $\chi \in S$ and its stabilizer $H=\{g\in G(k):{\chi }^g=\chi \}$. It has finite index because $G(k)$ permutes the finite set $S$. For every $n\in N(k)$ the functions $g\mapsto \chi (g^{-1}ng)$ and $g\mapsto \chi (n)$ are regular, so their equalizer is closed. Their intersection over all $n\in N(k)$ is closed; equality on these points is equality of characters as morphisms because the smooth group $N$ has schematically dense rational points [F5]. Thus this intersection is exactly $H$. The finite-index lemma [F4] gives $H=G(k)$, hence every $V_{\chi }$ is $G(k)$-stable.

Since $V$ is simple and $G$ is smooth over the algebraically closed field $k$, the stabilizer of the subspace $V_{\chi }$ is a closed subscheme of $G$ containing $G(k)$, hence equals $G$ by [F5]; thus $V_{\chi }$ is a nonzero $G$-subrepresentation of $V$, so $V=V_{\chi }$ and the sum in [F3] has a single term. Therefore each $n\in N(k)$ acts on $V$ as the homothety $\chi (n)$.

By [F2] every element of $N(k)=(DG)(k)$ is a product of commutators $[x,y]$ of elements of $G(k)$, hence acts on $V$ with determinant $1$. Since it acts as the homothety $\chi (n)$ with $d=\dim V$, its determinant is $\chi (n{)}^d$, so $\chi$ maps $N(k)$ into the group ${\mu }_d(k)$ of $d$-th roots of unity. As $N$ is smooth and connected and ${\mu }_d$ is finite, the image $\chi (N)$ is connected and finite, hence trivial; so $N$ acts trivially on $V$.

Consequently $V$ is a simple representation of the quotient $G\mathrm{/}N$, which is commutative by [F2]. By [F1] a simple finite-dimensional representation of the smooth commutative group $G\mathrm{/}N$ has dimension one; hence $\dim V=1$. This completes the induction: every simple representation of $G$ is one-dimensional, so $G$ is trigonalizable, and the equivalent flag formulation follows from Trigonalizable groups, invariant flags and embeddings into T_n.