Group-scheme action
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In
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, an action of a group scheme is a generalization of a
group action In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
to a
group scheme In mathematics, a group scheme is a type of object from Algebraic geometry, algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of Scheme (mathematics), schemes, and they generalize algebraic groups, in ...
. Precisely, given a group ''S''-scheme ''G'', a left action of ''G'' on an ''S''-scheme ''X'' is an ''S''-morphism :\sigma: G \times_S X \to X such that * (associativity) \sigma \circ (1_G \times \sigma) = \sigma \circ (m \times 1_X), where m: G \times_S G \to G is the group law, * (unitality) \sigma \circ (e \times 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 In mathematics, equivariance is a form of symmetry for functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry grou ...
between ''G''-schemes is a
morphism of schemes In algebraic geometry, a morphism of schemes generalizes a morphism of algebraic varieties just as a scheme generalizes an algebraic variety. It is, by definition, a morphism in the category of schemes. A morphism of algebraic stacks generalizes a ...
that intertwines the respective ''G''-actions. More generally, one can also consider (at least some special case of) an action of a
group functor In mathematics, a group functor is a group-valued functor on the category of commutative rings. Although it is typically viewed as a generalization of a group scheme, the notion itself involves no scheme theory. Because of this feature, some authors ...
: viewing ''G'' as a functor, an action is given as a natural transformation satisfying the conditions analogous to the above.In details, given a group-scheme action \sigma, for each morphism T \to S, \sigma determines a group action G(T) \times 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) \times X(T) \to X(T) and if those actions are compatible; i.e., they form a
natural transformation In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
, then, by the
Yoneda lemma In mathematics, the Yoneda lemma is arguably the most important result in category theory. It is an abstract result on functors of the type ''morphisms into a fixed object''. It is a vast generalisation of Cayley's theorem from group theory (view ...
, they determine a group-scheme action \sigma: G \times_S X \to X.
Alternatively, some authors study group action in the language of a
groupoid In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: *''Group'' with a partial functi ...
; a group-scheme action is then an example of a
groupoid scheme In category theory, a branch of mathematics, a groupoid object is both a generalization of a groupoid which is built on richer structures than sets, and a generalization of a group objects when the multiplication is only partially defined. Defin ...
.


Constructs

The usual constructs for a
group action In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
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 \times_S T \to X \times_S T is given as (\sigma \circ (1_G \times x), p_2). *The
orbit In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a p ...
of ''x'' is the image of the orbit map \sigma_x. *The stabilizer of ''x'' is the
fiber Fiber or fibre (from la, fibra, links=no) is a natural or artificial substance that is significantly longer than it is wide. Fibers are often used in the manufacture of other materials. The strongest engineering materials often incorporate ...
over \sigma_x of the map (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 In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equ ...
. There are several approaches to overcome this difficulty: *
Level structure In the mathematical subfield of graph theory a level structure of an undirected graph is a partition of the vertices into subsets that have the same distance from a given root vertex.. Definition and construction Given a connected graph ''G'' ...
- Perhaps the oldest, the approach replaces an object to classify by an object together with a level structure *
Geometric invariant theory In mathematics, geometric invariant theory (or GIT) is a method for constructing quotients by group actions in algebraic geometry, used to construct moduli spaces. It was developed by David Mumford in 1965, using ideas from the paper in clas ...
- 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 In mathematics, linearization is finding the linear approximation to a function at a given point. The linear approximation of a function is the first order Taylor expansion around the point of interest. In the study of dynamical systems, lineariz ...
. See also:
categorical quotient In algebraic geometry, given a category ''C'', a categorical quotient of an object ''X'' with action of a group ''G'' is a morphism \pi: X \to Y that :(i) is invariant; i.e., \pi \circ \sigma = \pi \circ p_2 where \sigma: G \times X \to X is the ...
,
GIT quotient In algebraic geometry, an affine GIT quotient, or affine geometric invariant theory quotient, of an affine scheme X = \operatorname A with an action by a group scheme ''G'' is the affine scheme \operatorname(A^G), the prime spectrum of the ring of ...
. *
Borel construction In mathematics, equivariant cohomology (or ''Borel cohomology'') is a cohomology theory from algebraic topology which applies to topological spaces with a ''group action''. It can be viewed as a common generalization of group cohomology and an or ...
- this is an approach essentially from algebraic topology; this approach requires one to work with an
infinite-dimensional space In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to dis ...
. *Analytic approach, the theory of
Teichmüller space In mathematics, the Teichmüller space T(S) of a (real) topological (or differential) surface S, is a space that parametrizes complex structures on S up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Teichmülle ...
*
Quotient stack In algebraic geometry, a quotient stack is a stack that parametrizes equivariant objects. Geometrically, it generalizes a quotient of a scheme or a variety by a group: a quotient variety, say, would be a coarse approximation of a quotient stack. T ...
- 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 In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category of sets and possess a notio ...
. So the problem shifts from the classification of orbits to that of
equivariant object In mathematics, equivariance is a form of symmetry for functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry group, ...
s.


See also

*
groupoid scheme In category theory, a branch of mathematics, a groupoid object is both a generalization of a groupoid which is built on richer structures than sets, and a generalization of a group objects when the multiplication is only partially defined. Defin ...
* Sumihiro's theorem *
equivariant sheaf In mathematics, given an Group-scheme action, action \sigma: G \times_S X \to X of a group scheme ''G'' on a scheme ''X'' over a base scheme ''S'', an equivariant sheaf ''F'' on ''X'' is a sheaf of sheaf of modules, \mathcal_X-modules together with ...
*
Borel fixed-point theorem In mathematics, the Borel fixed-point theorem is a fixed-point theorem in algebraic geometry generalizing the Lie–Kolchin theorem. The result was proved by . Statement If ''G'' is a connected, solvable, linear algebraic group acting regularly ...


References

* {{algebraic-geometry-stub Algebraic geometry