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In mathematics, a group scheme is a type of object from algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they generalize
algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Ma ...
s, in the sense that all algebraic groups have group scheme structure, but group schemes are not necessarily connected, smooth, or defined over a field. This extra generality allows one to study richer infinitesimal structures, and this can help one to understand and answer questions of arithmetic significance. The
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...
of group schemes is somewhat better behaved than that of group varieties, since all homomorphisms have
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learn ...
s, and there is a well-behaved deformation theory. Group schemes that are not algebraic groups play a significant role in
arithmetic geometry In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic varieties. ...
and
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
, since they come up in contexts of
Galois representation In mathematics, a Galois module is a ''G''-module, with ''G'' being the Galois group of some extension of fields. The term Galois representation is frequently used when the ''G''-module is a vector space over a field or a free module over a ring ...
s and
moduli problem In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such s ...
s. The initial development of the theory of group schemes was due to Alexander Grothendieck,
Michel Raynaud Michel Raynaud (; 16 June 1938 – 10 March 2018 Décès de Michel Raynaud
So ...
and
Michel Demazure Michel Demazure (; born 2 March 1937) is a French mathematician. He made contributions in the fields of abstract algebra, algebraic geometry, and computer vision, and participated in the Nicolas Bourbaki collective. He has also been president ...
in the early 1960s.


Definition

A group scheme is a
group object In category theory, a branch of mathematics, group objects are certain generalizations of groups that are built on more complicated structures than sets. A typical example of a group object is a topological group, a group whose underlying set is ...
in a category of schemes that has fiber products and some final object ''S''. That is, it is an ''S''-scheme ''G'' equipped with one of the equivalent sets of data * a triple of morphisms μ: ''G'' ×S ''G'' → ''G'', e: ''S'' → ''G'', and ι: ''G'' → ''G'', satisfying the usual compatibilities of groups (namely associativity of μ, identity, and inverse axioms) * a functor from schemes over ''S'' to the
category of groups In mathematics, the category Grp (or Gp) has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category. The study of this category is known as group theory. Relation to other categories There a ...
, such that composition with the
forgetful functor In mathematics, in the area of category theory, a forgetful functor (also known as a stripping functor) 'forgets' or drops some or all of the input's structure or properties 'before' mapping to the output. For an algebraic structure of a given sign ...
to sets is equivalent to the presheaf corresponding to ''G'' under the
Yoneda embedding 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 (vie ...
. (See also:
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 author ...
.) A homomorphism of group schemes is a map of schemes that respects multiplication. This can be precisely phrased either by saying that a map ''f'' satisfies the equation ''f''μ = μ(''f'' × ''f''), or by saying that ''f'' is 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 ...
of functors from schemes to groups (rather than just sets). A left action of a group scheme ''G'' on a scheme ''X'' is a morphism ''G'' ×S ''X''→ ''X'' that induces a left
action Action may refer to: * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video game Film * Action film, a genre of film * ''Action'' (1921 film), a film by John Ford * ''Action'' (1980 fil ...
of the group ''G''(''T'') on the set ''X''(''T'') for any ''S''-scheme ''T''. Right actions are defined similarly. Any group scheme admits natural left and right actions on its underlying scheme by multiplication and
conjugation Conjugation or conjugate may refer to: Linguistics * Grammatical conjugation, the modification of a verb from its basic form * Emotive conjugation or Russell's conjugation, the use of loaded language Mathematics * Complex conjugation, the chang ...
. Conjugation is an action by automorphisms, i.e., it commutes with the group structure, and this induces linear actions on naturally derived objects, such as its Lie algebra, and the algebra of left-invariant differential operators. An ''S''-group scheme ''G'' is commutative if the group ''G''(''T'') is an abelian group for all ''S''-schemes ''T''. There are several other equivalent conditions, such as conjugation inducing a trivial action, or inversion map ι being a group scheme automorphism.


Constructions

* Given a group ''G'', one can form the constant group scheme ''G''''S''. As a scheme, it is a disjoint union of copies of ''S'', and by choosing an identification of these copies with elements of ''G'', one can define the multiplication, unit, and inverse maps by transport of structure. As a functor, it takes any ''S''-scheme ''T'' to a product of copies of the group ''G'', where the number of copies is equal to the number of connected components of ''T''. ''G''''S'' is affine over ''S'' if and only if ''G'' is a finite group. However, one can take a projective limit of finite constant group schemes to get profinite group schemes, which appear in the study of fundamental groups and Galois representations or in the theory of the
fundamental group scheme In mathematics, the fundamental group scheme is a group scheme canonically attached to a scheme over a Dedekind scheme (e.g. the spectrum of a field or the spectrum of a discrete valuation ring). It is a generalisation of the étale fundamental g ...
, and these are affine of infinite type. More generally, by taking a locally constant sheaf of groups on ''S'', one obtains a locally constant group scheme, for which monodromy on the base can induce non-trivial automorphisms on the fibers. * The existence of fiber products of schemes allows one to make several constructions. Finite direct products of group schemes have a canonical group scheme structure. Given an action of one group scheme on another by automorphisms, one can form semidirect products by following the usual set-theoretic construction. Kernels of group scheme homomorphisms are group schemes, by taking a fiber product over the unit map from the base. Base change sends group schemes to group schemes. * Group schemes can be formed from smaller group schemes by taking
restriction of scalars In algebra, given a ring homomorphism f: R \to S, there are three ways to change the coefficient ring of a module; namely, for a left ''R''-module ''M'' and a left ''S''-module ''N'', *f_! M = S\otimes_R M, the induced module. *f_* M = \operator ...
with respect to some morphism of base schemes, although one needs finiteness conditions to be satisfied to ensure representability of the resulting functor. When this morphism is along a finite extension of fields, it is known as Weil restriction. * For any abelian group ''A'', one can form the corresponding diagonalizable group ''D''(''A''), defined as a functor by setting ''D''(''A'')(''T'') to be the set of abelian group homomorphisms from ''A'' to invertible global sections of ''O''T for each ''S''-scheme ''T''. If ''S'' is affine, ''D''(''A'') can be formed as the spectrum of a group ring. More generally, one can form groups of multiplicative type by letting ''A'' be a non-constant sheaf of abelian groups on ''S''. * For a subgroup scheme ''H'' of a group scheme ''G'', the functor that takes an ''S''-scheme ''T'' to ''G''(''T'')/''H''(''T'') is in general not a sheaf, and even its sheafification is in general not representable as a scheme. However, if ''H'' is finite, flat, and closed in ''G'', then the quotient is representable, and admits a canonical left ''G''-action by translation. If the restriction of this action to ''H'' is trivial, then ''H'' is said to be normal, and the quotient scheme admits a natural group law. Representability holds in many other cases, such as when ''H'' is closed in ''G'' and both are affine.


Examples

* The multiplicative group Gm has the punctured affine line as its underlying scheme, and as a functor, it sends an ''S''-scheme ''T'' to the multiplicative group of invertible global sections of the structure sheaf. It can be described as the diagonalizable group ''D''(Z) associated to the integers. Over an affine base such as Spec ''A'', it is the spectrum of the ring ''A'' 'x'',''y''(''xy'' − 1), which is also written ''A'' 'x'', ''x''−1 The unit map is given by sending ''x'' to one, multiplication is given by sending ''x'' to ''x'' ⊗ ''x'', and the inverse is given by sending ''x'' to ''x''−1.
Algebraic tori Algebraic may refer to any subject related to algebra in mathematics and related branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings. Algebraic may also refer to: * Algebraic data type, a dat ...
form an important class of commutative group schemes, defined either by the property of being locally on ''S'' a product of copies of Gm, or as groups of multiplicative type associated to finitely generated free abelian groups. * The general linear group ''GL''''n'' is an affine algebraic variety that can be viewed as the multiplicative group of the ''n'' by ''n'' matrix ring variety. As a functor, it sends an ''S''-scheme ''T'' to the group of invertible ''n'' by ''n'' matrices whose entries are global sections of ''T''. Over an affine base, one can construct it as a quotient of a polynomial ring in ''n''2 + 1 variables by an ideal encoding the invertibility of the determinant. Alternatively, it can be constructed using 2''n''2 variables, with relations describing an ordered pair of mutually inverse matrices. * For any positive integer ''n'', the group μn is the kernel of the ''n''th power map from Gm to itself. As a functor, it sends any ''S''-scheme ''T'' to the group of global sections ''f'' of ''T'' such that ''f''n = 1. Over an affine base such as Spec ''A'', it is the spectrum of ''A'' (''x''''n''−1). If ''n'' is not invertible in the base, then this scheme is not smooth. In particular, over a field of characteristic ''p'', μp is not smooth. * The additive group Ga has the affine line A1 as its underlying scheme. As a functor, it sends any ''S''-scheme ''T'' to the underlying additive group of global sections of the structure sheaf. Over an affine base such as Spec ''A'', it is the spectrum of the polynomial ring ''A'' 'x'' The unit map is given by sending ''x'' to zero, the multiplication is given by sending ''x'' to 1 ⊗ ''x'' + ''x'' ⊗ 1, and the inverse is given by sending ''x'' to −''x''. * If ''p'' = 0 in ''S'' for some prime number ''p'', then the taking of ''p''th powers induces an endomorphism of Ga, and the kernel is the group scheme αp. Over an affine base such as Spec ''A'', it is the spectrum of ''A'' (''x''p). * The automorphism group of the affine line is isomorphic to the semidirect product of Ga by Gm, where the additive group acts by translations, and the multiplicative group acts by dilations. The subgroup fixing a chosen basepoint is isomorphic to the multiplicative group, and taking the basepoint to be the identity of an additive group structure identifies Gm with the automorphism group of Ga. * A smooth genus one curve with a marked point (i.e., an
elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
) has a unique group scheme structure with that point as the identity. Unlike the previous positive-dimensional examples, elliptic curves are projective (in particular proper).


Basic properties

Suppose that ''G'' is a group scheme of finite type over a field ''k''. Let ''G''0 be the connected component of the identity, i.e., the maximal connected subgroup scheme. Then ''G'' is an extension of a finite étale group scheme by ''G''0. ''G'' has a unique maximal reduced subscheme ''G''red, and if ''k'' is perfect, then ''G''red is a smooth group variety that is a subgroup scheme of ''G''. The quotient scheme is the spectrum of a local ring of finite rank. Any affine group scheme is the
spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors ...
of a commutative Hopf algebra (over a base ''S'', this is given by the relative spectrum of an ''O''S-algebra). The multiplication, unit, and inverse maps of the group scheme are given by the comultiplication, counit, and antipode structures in the Hopf algebra. The unit and multiplication structures in the Hopf algebra are intrinsic to the underlying scheme. For an arbitrary group scheme ''G'', the ring of global sections also has a commutative Hopf algebra structure, and by taking its spectrum, one obtains the maximal affine quotient group. Affine group varieties are known as linear algebraic groups, since they can be embedded as subgroups of general linear groups. Complete connected group schemes are in some sense opposite to affine group schemes, since the completeness implies all global sections are exactly those pulled back from the base, and in particular, they have no nontrivial maps to affine schemes. Any complete group variety (variety here meaning reduced and geometrically irreducible separated scheme of finite type over a field) is automatically commutative, by an argument involving the action of conjugation on jet spaces of the identity. Complete group varieties are called
abelian varieties In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a Algebraic variety#Projective variety, projective algebraic variety that is also an algebraic group, i.e., has a group law th ...
. This generalizes to the notion of abelian scheme; a group scheme ''G'' over a base ''S'' is abelian if the structural morphism from ''G'' to ''S'' is proper and smooth with geometrically connected fibers. They are automatically projective, and they have many applications, e.g., in geometric
class field theory In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field. Hilbert is credit ...
and throughout algebraic geometry. A complete group scheme over a field need not be commutative, however; for example, any finite group scheme is complete.


Finite flat group schemes

A group scheme ''G'' over a noetherian scheme ''S'' is finite and flat if and only if ''O''''G'' is a locally free ''O''''S''-module of finite rank. The rank is a locally constant function on ''S'', and is called the order of ''G''. The order of a constant group scheme is equal to the order of the corresponding group, and in general, order behaves well with respect to base change and finite flat
restriction of scalars In algebra, given a ring homomorphism f: R \to S, there are three ways to change the coefficient ring of a module; namely, for a left ''R''-module ''M'' and a left ''S''-module ''N'', *f_! M = S\otimes_R M, the induced module. *f_* M = \operator ...
. Among the finite flat group schemes, the constants (cf. example above) form a special class, and over an algebraically closed field of characteristic zero, the category of finite groups is equivalent to the category of constant finite group schemes. Over bases with positive characteristic or more arithmetic structure, additional isomorphism types exist. For example, if 2 is invertible over the base, all group schemes of order 2 are constant, but over the 2-adic integers, μ2 is non-constant, because the special fiber isn't smooth. There exist sequences of highly ramified 2-adic rings over which the number of isomorphism types of group schemes of order 2 grows arbitrarily large. More detailed analysis of commutative finite flat group schemes over ''p''-adic rings can be found in Raynaud's work on prolongations. Commutative finite flat group schemes often occur in nature as subgroup schemes of abelian and semi-abelian varieties, and in positive or mixed characteristic, they can capture a lot of information about the ambient variety. For example, the ''p''-torsion of an elliptic curve in characteristic zero is locally isomorphic to the constant elementary abelian group scheme of order ''p''2, but over Fp, it is a finite flat group scheme of order ''p''2 that has either ''p'' connected components (if the curve is ordinary) or one connected component (if the curve is
supersingular In mathematics, a supersingular variety is (usually) a smooth projective variety in nonzero characteristic such that for all ''n'' the slopes of the Newton polygon of the ''n''th crystalline cohomology are all ''n''/2 . For special classes o ...
). If we consider a family of elliptic curves, the ''p''-torsion forms a finite flat group scheme over the parametrizing space, and the supersingular locus is where the fibers are connected. This merging of connected components can be studied in fine detail by passing from a modular scheme to a
rigid analytic space In mathematics, a rigid analytic space is an analogue of a complex analytic space over a nonarchimedean field. Such spaces were introduced by John Tate in 1962, as an outgrowth of his work on uniformizing ''p''-adic elliptic curves with bad red ...
, where supersingular points are replaced by discs of positive radius.


Cartier duality

Cartier duality is a scheme-theoretic analogue of
Pontryagin duality In mathematics, Pontryagin duality is a duality (mathematics), duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numb ...
taking finite commutative group schemes to finite commutative group schemes.


Dieudonné modules

Finite flat commutative group schemes over a perfect field ''k'' of positive characteristic ''p'' can be studied by transferring their geometric structure to a (semi-)linear-algebraic setting. The basic object is the Dieudonné ring ''D'' = ''W''(''k'')/(''FV'' − ''p''), which is a quotient of the ring of noncommutative polynomials, with coefficients in
Witt vectors In mathematics, a Witt vector is an infinite sequence of elements of a commutative ring. Ernst Witt showed how to put a ring structure on the set of Witt vectors, in such a way that the ring of Witt vectors W(\mathbb_p) over the finite field of ord ...
of ''k''. ''F'' and ''V'' are the Frobenius and Verschiebung operators, and they may act nontrivially on the Witt vectors. Dieudonne and Cartier constructed an antiequivalence of categories between finite commutative group schemes over ''k'' of order a power of "p" and modules over ''D'' with finite ''W''(''k'')-length. The Dieudonné module functor in one direction is given by homomorphisms into the abelian sheaf ''CW'' of Witt co-vectors. This sheaf is more or less dual to the sheaf of Witt vectors (which is in fact representable by a group scheme), since it is constructed by taking a direct limit of finite length Witt vectors under successive Verschiebung maps ''V'': ''W''n → ''W''n+1, and then completing. Many properties of commutative group schemes can be seen by examining the corresponding Dieudonné modules, e.g., connected ''p''-group schemes correspond to ''D''-modules for which ''F'' is nilpotent, and étale group schemes correspond to modules for which ''F'' is an isomorphism. Dieudonné theory exists in a somewhat more general setting than finite flat groups over a field. Oda's 1967 thesis gave a connection between Dieudonné modules and the first de Rham cohomology of abelian varieties, and at about the same time, Grothendieck suggested that there should be a crystalline version of the theory that could be used to analyze ''p''-divisible groups. Galois actions on the group schemes transfer through the equivalences of categories, and the associated deformation theory of Galois representations was used in Wiles's work on the Shimura–Taniyama conjecture.


See also

*
Fundamental group scheme In mathematics, the fundamental group scheme is a group scheme canonically attached to a scheme over a Dedekind scheme (e.g. the spectrum of a field or the spectrum of a discrete valuation ring). It is a generalisation of the étale fundamental g ...
*
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 ...
*
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 ...
* Groupoid scheme *
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 :\sigma: G \times_S X \to ...
*
Group-stack In algebraic geometry, a group-stack is an algebraic stack whose categories of points have group structures or even groupoid structures in a compatible way. It generalizes a group scheme, which is a scheme whose sets of points have group structures ...
*
Invariant theory Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit descri ...
*
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. Th ...


References

* * * * *Berthelot, Breen, Messing ''Théorie de Dieudonné Crystalline II'' *Laumon, ''Transformation de Fourier généralisée'' * * * John Tate, ''Finite flat group schemes'', from ''Modular Forms and Fermat's Last Theorem'' * {{Authority control Algebraic groups Scheme theory Hopf algebras Duality theories