Group-scheme action

In algebraic geometry, an action of a group scheme is a generalization of a group action to a group scheme. Precisely, given a group "S"-scheme "G", a left action of "G" on an "S"-scheme "X" is an "S"-morphism
such that
A right action of "G" on "X is defined analogously. A scheme equipped with a left or right action of a group scheme "G" is called a G"-scheme. An equivariant morphism between "G"-schemes is a morphism of schemes that intertwines the respective "G"-actions.
More generally, one can also consider (at least some special case of) an action of a group functor: viewing "G" as a functor, an action is given as a natural transformation satisfying the conditions analogous to the above. Alternatively, some authors study group action in the language of a groupoid; a group-scheme action is then an example of a groupoid scheme.
Constructs.
The usual constructs for a group action such as orbits generalize to a group-scheme action. Let formula_6 be a given group-scheme action as above.
Problem of constructing a quotient.
Unlike a set-theoretic group action, there is no straightforward way to construct a quotient for a group-scheme action. One exception is the case when the action is free, the case of a principal fiber bundle.
There are several approaches to overcome this difficulty:
Depending on applications, another approach would be to shift the focus away from a space then onto stuff on a space; e.g., topos. So the problem shifts from the classification of orbits to that of equivariant objects.