# Rosenlicht's theorem and rationality questions

Let $G$ be a connected algebraic group over an algebraically closed field $\overline{k}$ acting on an irreducible variety $X$. A geometric quotient is a morphism of varieties $\pi: X \rightarrow X/\sim$ which on closed points (that is, as a morphism of classical varieties) satisfy the following:

(i): $\pi$ is a surjective open map, and the fibres are exactly the $G$-orbits of $X$.

(ii): for any open set $U$ of $X/\sim$, the ring homomorphism $\pi^{\ast}: \overline{k}[U] \rightarrow \overline{k}[\pi^{-1}U]$ is an isomorphism onto the $G$-fixed points of codomain.

Rosenlicht's theorem says that there exists a $G$-stable open subset of $X$ for which the geometric quotient exists.

Is there any generalization of Rosenlicht's theorem for when $G$ and $X$ are defined over an arbitrary field? The case I'm interested in is when $G$ and $X$ are geometrically connected subgroups of upper triangular unipotent matrices over a $p$-adic field $F$ (so all orbits are closed), and $X$ is normalized by $G$.

• Assume that $G$ and $X$ are defined over a perfect field $k$. You can choose a $G$-stable open subset $U$ of $X$ (for which the geometric quotient exists) defined over $k$. It seems that your geometric quotient $U/\sim$ is has no automorphisms (compatible with the identity automorphism of $U$). If $U/\sim$ is quasi-projective, you can construct a $k$-model of it by Galois descent. (I hope all this is correct....) Jan 18, 2018 at 9:24
• @MikhailBorovoi. You need to make sure that the $G$-stable open subset $U$ descends, or equivalently, that its complement descends. Since there are finitely many coefficients of the finitely many defining equations of the closed complement, the open subset $U$ descends to some finite field extension $L/k$. Since $k$ is perfect, this field extension is Galois. Now you can replace $U$ by the intersection of the finitely many Galois conjugates of $U$. This $G$-stable open subset descends. Jan 18, 2018 at 13:43
• @JasonStarr: Yes, this is what I meant, the intersection of finitely many Galois conjugates of $U$. I was in hurry and could not write it down. Jan 18, 2018 at 16:30
• @D_S: It would help to give a reference for Rosenlicht's result. He wrote quite a few papers on structural questions for algebraic groups and their actions, often in language which is now getting archaic but also very often dealing with fields of definition. Jan 18, 2018 at 17:06
• @D_S: An unrelated question is why you've included the tag 'reductive-groups'? Probably 'group-actions' would be a better choice if you need another tag. Jan 18, 2018 at 17:12