# 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

- (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

**. An equivariant morphism between**

*G*-scheme*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 .
- 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 apppraoch 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.

## References

- 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 .

- Mumford, David; Fogarty, J.; Kirwan, F. (1994).
*Geometric invariant theory*. Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)].**34**(3rd ed.). Berlin, New York: Springer-Verlag. ISBN 978-3-540-56963-3. MR 1304906.