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

${\displaystyle \sigma :G\times _{S}X\to X}$

such that

• (associativity) ${\displaystyle \sigma \circ (1_{G}\times \sigma )=\sigma \circ (m\times 1_{X})}$, where ${\displaystyle m:G\times _{S}G\to G}$ is the group law,
• (unitality) ${\displaystyle \sigma \circ (e\times 1_{X})=1_{X}}$, where ${\displaystyle e:S\to G}$ 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. ^[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 ${\displaystyle \sigma }$ be a given group-scheme action as above.

• Given a T-valued point ${\displaystyle x:T\to X}$, the orbit map ${\displaystyle \sigma _{x}:G\times _{S}T\to X\times _{S}T}$ is given as ${\displaystyle (\sigma \circ (1_{G}\times x),p_{2})}$.
• The orbit of x is the image of the orbit map ${\displaystyle \sigma _{x}}$.
• The stabilizer of x is the fiber over ${\displaystyle \sigma _{x}}$ of the map ${\displaystyle (x,1_{T}):T\to X\times _{S}T.}$

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

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

References

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

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