“Group-scheme action”的意思、由来-开放百科全书

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Group-scheme action

释义

Constructs
Problem of constructing a quotient
See also
References

In algebraic geometry, a 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

such that

(associativity) , where is the group law,
(unitality) , where 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 be a given group-scheme action as above.

Given a T-valued point , the orbit map is given as .
The orbit of x is the image of the orbit map .
The stabilizer of x is the fiber over of the map

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 apppraoch 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.

References

1. ^In details, given a group-scheme action

, for each morphism

determines a group action

; i.e., the group

acts on the set of T-points

. Conversely, if for each

, there is a group action

and if those actions are compatible; i.e., they form a natural transformation, then, by the Yoneda lemma, they determine a group-scheme action

{{Cite book| last1=Mumford | first1=David | author1-link=David Mumford | last2=Fogarty | first2=J. | last3=Kirwan | first3=F. | author3-link=Frances Kirwan | title=Geometric invariant theory | publisher=Springer-Verlag | location=Berlin, New York | edition=3rd | series=Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)] | isbn=978-3-540-56963-3 |mr=1304906 | year=1994 | volume=34}}

1 : Algebraic geometry

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Constructs

Problem of constructing a quotient

See also

References