Quotient (algebraic Geometry)

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.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 $\sigma: G \times_S X \to X$. 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.

See also

*groupoid scheme
*Sumihiro's theorem
*equivariant sheaf
*Borel fixed-point theorem

References

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{{algebraic-geometry-stub

Algebraic geometry