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
σ : G × S X → X {\displaystyle \sigma :G\times _{S}X\to X}such that
- (associativity) σ ∘ ( 1 G × σ ) = σ ∘ ( m × 1 X ) {\displaystyle \sigma \circ (1_{G}\times \sigma )=\sigma \circ (m\times 1_{X})} , where m : G × S G → G {\displaystyle m:G\times _{S}G\to G} is the group law,
- (unitality) σ ∘ ( e × 1 X ) = 1 X {\displaystyle \sigma \circ (e\times 1_{X})=1_{X}} , where e : S → G {\displaystyle 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 σ {\displaystyle \sigma } be a given group-scheme action as above.
- Given a T-valued point x : T → X {\displaystyle x:T\to X} , the orbit map σ x : G × S T → X × S T {\displaystyle \sigma _{x}:G\times _{S}T\to X\times _{S}T} is given as ( σ ∘ ( 1 G × x ) , p 2 ) {\displaystyle (\sigma \circ (1_{G}\times x),p_{2})} .
- The orbit of x is the image of the orbit map σ x {\displaystyle \sigma _{x}} .
- The stabilizer of x is the fiber over σ x {\displaystyle \sigma _{x}} of the map ( x , 1 T ) : T → X × S T . {\displaystyle (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
- 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.
References
In details, given a group-scheme action σ {\displaystyle \sigma } , for each morphism T → S {\displaystyle T\to S} , σ {\displaystyle \sigma } determines a group action G ( T ) × X ( T ) → X ( T ) {\displaystyle G(T)\times X(T)\to X(T)} ; i.e., the group G ( T ) {\displaystyle G(T)} acts on the set of T-points X ( T ) {\displaystyle X(T)} . Conversely, if for each T → S {\displaystyle T\to S} , there is a group action σ T : G ( T ) × X ( T ) → X ( T ) {\displaystyle \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 σ : G × S X → X {\displaystyle \sigma :G\times _{S}X\to X} . /wiki/Natural_transformation ↩