Question

A Lie group can (often) be recovered as the $\mathbb{R}$-points of a group scheme. I am wondering if this parallelism carries over to proper actions.

In particular, let $G$ be a Lie group acting on a manifold $M$. Let $\underline G$ be the associated group scheme acting on the scheme $\underline M$, where the $\mathbb{R}$-points of $\underline M$ can be identified with $M$.

Could it be true that the map $G\times M\to M\times M$ is proper (in the topological sense) if and only if $\underline G\times \underline M\to \underline M\times \underline M$ is a proper morphism of schemes?

Answer 1

A morphism of schemes over $\mathbb C$ is proper iff the map on topological spaces is proper, so the whole thing works for complex points if the fibred products are over $\mathbb C$ as well. Over the reals it goes wrong as the following example shows: Let $G$ be the group scheme $SO(2)$ and let $M$ be a single point. The real points of $G$ form a compact group, so the map $G({\mathbb R})\times pt\to pt\times pt$ is proper, but the corresponding map of schemes is not, as $G$ is affine of positive dimension.