# 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

$\sigma :G\times _{S}X\to X$ such that

• (associativity) $\sigma \circ (1_{G}\times \sigma )=\sigma \circ (m\times 1_{X})$ , where $m:G\times _{S}G\to G$ is the group law,
• (unitality) $\sigma \circ (e\times 1_{X})=1_{X}$ , where $e:S\to G$ 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. 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 $\sigma$ be a given group-scheme action as above.

• Given a T-valued point $x:T\to X$ , the orbit map $\sigma _{x}:G\times _{S}T\to X\times _{S}T$ is given as $(\sigma \circ (1_{G}\times x),p_{2})$ .
• The orbit of x is the image of the orbit map $\sigma _{x}$ .
• The stabilizer of x is the fiber over $\sigma _{x}$ of the map $(x,1_{T}):T\to X\times _{S}T.$ ## 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 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.