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 (associativity) , where is the group law, (unitality) , where is the identity section of G.
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.In details, given a group-scheme action , for each morphism , determines a group action ; i.e., the group acts on the set of T-points . Conversely, if for each , there is a group action and if those actions are compatible; i.e., they form a natural transformation, then, by the Yoneda lemma, they determine a group-scheme action . 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 be a given group-scheme action as above.
Given a T-valued point , the orbit map is given as . The orbit of x is the image of the orbit map . The stabilizer of x is the fiber over of the map
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: Level structure - Perhaps the oldest, the approach replaces an object to classify by an object together with a level structure Geometric invariant theory - throw away bad orbits and then take a quotient. The drawback is that there is no canonical way to introduce the notion of "bad orbits"; the notion depends on a choice of linearization. See also: categorical quotient, GIT quotient. Borel construction - this is an approach essentially from algebraic topology; this approach requires one to work with an infinite-dimensional space. Analytic approach, the theory of Teichmüller space Quotient stack - in a sense, this is the ultimate answer to the problem. Roughly, a "quotient prestack" is the category of orbits and one stackify (i.e., the introduction of the notion of a torsor) it to get a quotient stack.
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.
See also groupoid scheme Sumihiro's theorem equivariant sheaf Borel fixed-point theorem
References