Английская Википедия: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
- <math>\sigma: G \times_S X \to X</math>
such that
- (associativity) <math>\sigma \circ (1_G \times \sigma) = \sigma \circ (m \times 1_X)</math>, where <math>m: G \times_S G \to G</math> is the group law,
- (unitality) <math>\sigma \circ (e \times 1_X) = 1_X</math>, where <math>e: S \to G</math> 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 <math>\sigma</math> be a given group-scheme action as above.
- Given a T-valued point <math>x: T \to X</math>, the orbit map <math>\sigma_x: G \times_S T \to X \times_S T</math> is given as <math>(\sigma \circ (1_G \times x), p_2)</math>.
- The orbit of x is the image of the orbit map <math>\sigma_x</math>.
- The stabilizer of x is the fiber over <math>\sigma_x</math> of the map <math>(x, 1_T): T \to X \times_S T.</math>
Problem of constructing a quotient
Шаблон:Expand section 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
References
Шаблон:Algebraic-geometry-stub
- ↑ In details, given a group-scheme action <math>\sigma</math>, for each morphism <math>T \to S</math>, <math>\sigma</math> determines a group action <math>G(T) \times X(T) \to X(T)</math>; i.e., the group <math>G(T)</math> acts on the set of T-points <math>X(T)</math>. Conversely, if for each <math>T \to S</math>, there is a group action <math>\sigma_T: G(T) \times X(T) \to X(T)</math> and if those actions are compatible; i.e., they form a natural transformation, then, by the Yoneda lemma, they determine a group-scheme action <math>\sigma: G \times_S X \to X</math>.
