- Source: 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
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
groupoid scheme
Sumihiro's theorem
equivariant sheaf
Borel fixed-point theorem
References
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.
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