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.[1] 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 .
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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
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.
^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 .