Group-scheme action explained

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 x _{S} X \to X such that

• (associativity)

\sigma \circ ( 1 _{G} x \sigma) = \sigma \circ (m x 1 _{X)} , where m: G x _{S} G \to G is the group law,

• (unitality)

\sigma \circ (e x 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 x _{S} T \to X x _{S} T is given as (\sigma \circ ( 1 _{G} x 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 x _{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

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

Notes and References

1. In details, given a group-scheme action \sigma , for each morphism T \to S , \sigma determines a group action G(T) x 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) x 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 x _{S} X \to X .