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.[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 $\sigma$ 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:

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.

References

In details, given a group-scheme action $\sigma$ , for each morphism $T\to S$ , $\sigma$ determines a group action $G(T)\times X(T)\to X(T)$ ; i.e., the group $G(T)$ acts on the set of T-points $X(T)$ . Conversely, if for each $T\to S$ , there is a group action $\sigma _{T}:G(T)\times X(T)\to X(T)$ and if those actions are compatible; i.e., they form a natural transformation, then, by the Yoneda lemma, they determine a group-scheme action $\sigma :G\times _{S}X\to X$ .

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

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[1] In details, given a group-scheme action $\sigma$ , for each morphism $T\to S$ , $\sigma$ determines a group action $G(T)\times X(T)\to X(T)$ ; i.e., the group $G(T)$ acts on the set of T-points $X(T)$ . Conversely, if for each $T\to S$ , there is a group action $\sigma _{T}:G(T)\times X(T)\to X(T)$ and if those actions are compatible; i.e., they form a natural transformation, then, by the Yoneda lemma, they determine a group-scheme action $\sigma :G\times _{S}X\to X$ .

Group-scheme action

Constructs

Problem of constructing a quotient

See also

References