I am relatively inexperienced with schemes and I am having trouble finding references to results specifying when the projection morphism from a fiber product with a group scheme is étale. More precisely:
Let X be an S-scheme, G a group scheme acting on X by the map $G \times_{S}X \xrightarrow{\rho}X$. Under which conditions is the projection map on X étale?
Context:
Topological action groupoid by discrete group action is étale:
In the category of topological spaces, we say an internal groupoid G is étale if the source morphism from arrows to objects is a local homeomorphism. If $\Gamma$ is a discrete group acting on a topological space $X$, then projection $\Gamma \times X \xrightarrow{\pi_{2}} X$ is a local homeomorphism. We will denote the action groupoid of the action of $\Gamma$ on $X$ by $X\rtimes \Gamma$ where:
- $Obj(X\rtimes \Gamma) = X$,
- $Mor(X\rtimes \Gamma) = \Gamma \times X$ and,
- with $x, x' \in X = Obj(X\rtimes \Gamma)$, $Hom(x, x') = \{(g, x) \in \Gamma \times X \mid gx = x'\}$.
Under the same hypotheses, the groupoid source morphism $s$ is equal to $\pi_2$ and $X\rtimes \Gamma$ is an étale groupoid. In particular, arrows are an étale space for objects in this groupoid. Similar results exist for Lie groupoids in the category of smooth manifolds, where the notion of étale groupoid originates from.
Now, say I have some group scheme action $G\times_SX \rightarrow X$. From what I have understood so far, the algebraic equivalent of local homeomorphisms and étale spaces are étale morphisms and étale coverings. I suppose the rigorous description of this is https://ncatlab.org/nlab/show/%C3%A9tale+space#definition_for_arbitrary_toposes but I do not have the skills at the moment to tackle this. Following comments in https://ncatlab.org/nlab/show/categorification+via+groupoid+schemes, I understand that we can define an internal action groupoid in the category of schemes in a similar fashion, with $Obj(X\rtimes G) = X$ and $Mor(X\rtimes G) = G\times_SX.$ I'd like to have an analogous result to the previous one on topological spaces, thus describing the (arrows) fiber product as an étale cover (with possibly a single component) over X.
Finally, https://mathoverflow.net/questions/21981/stacks-vs-groupoids alludes to the fact that stacks might be the more natural object for algebraic purposes. Should I push for an understanding of quotient stacks for a more appropriate understanding of these questions?