Editing Group scheme

{{Group theory sidebar |Basics}}

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 [[Scheme (mathematics)|scheme]]s, and they generalize [[algebraic group]]s, in the sense that all algebraic groups have group scheme structure, but group schemes are not necessarily [[Connected space|connected]], [[Smooth scheme|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 (mathematics)|category]] of group schemes is somewhat better behaved than that of [[Group variety|group varieties]], since all homomorphisms have [[Kernel (category theory)|kernel]]s, and there is a well-behaved [[deformation theory]].  Group schemes that are not algebraic groups play a significant role in [[arithmetic geometry]] and [[algebraic topology]], since they come up in contexts of [[Galois representation]]s and [[moduli problem]]s.  The initial development of the theory of group schemes was due to [[Alexander Grothendieck]], [[Michel Raynaud]] and [[Michel Demazure]] in the early 1960s.

== Definition ==

A group scheme is a [[group object]] in a [[category of schemes]] that has [[fiber product of schemes|fiber product]]s 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'' ×<sub>S</sub> ''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]], such that composition with the [[forgetful functor]] to [[Set (mathematics)|sets]] is equivalent to the presheaf corresponding to ''G'' under the [[Yoneda lemma|Yoneda embedding]]. (See also: [[group functor]].)

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]] of functors from schemes to groups (rather than just sets).

A [[group-scheme action|left action of a group scheme]] ''G'' on a scheme ''X'' is a morphism ''G'' ×<sub>S</sub> ''X'' → ''X'' that induces a left [[Group action (mathematics)|action]] 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 [[inner automorphism|conjugation]].  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''<sub>''S''</sub>.  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''<sub>''S''</sub> 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]], 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 product of schemes|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 product]]s 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]] with respect to some morphism of base schemes, although one needs finiteness conditions to be satisfied to ensure [[representable functor|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''<sub>T</sub> 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.<ref>{{Citation | last1=Raynaud | first1=Michel | author1-link=Michel Raynaud | title=Passage au quotient par une relation d'équivalence plate | publisher=[[Springer-Verlag]] | location=Berlin, New York |mr=0232781 | year=1967}}</ref>

== Examples ==

* The multiplicative group '''G'''<sub>m</sub> 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''&nbsp;&minus;&nbsp;1), which is also written ''A''[''x'', ''x''<sup>&minus;1</sup>].  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''<sup>&minus;1</sup>.  [[algebraic torus|Algebraic tori]] form an important class of commutative group schemes, defined either by the property of being locally on ''S'' a product of copies of '''G'''<sub>m</sub>, or as groups of multiplicative type associated to finitely generated [[free abelian group]]s.
* The general linear group ''GL''<sub>''n''</sub> 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''<sup>2</sup> + 1 variables by an ideal encoding the invertibility of the determinant.  Alternatively, it can be constructed using 2''n''<sup>2</sup> variables, with relations describing an ordered pair of mutually inverse matrices.
* For any positive integer ''n'', the group μ<sub>n</sub> is the kernel of the ''n''th power map from '''G'''<sub>m</sub> to itself.  As a functor, it sends any ''S''-scheme ''T'' to the group of global sections ''f'' of ''T'' such that ''f''<sup>n</sup> = 1.  Over an affine base such as Spec ''A'', it is the spectrum of ''A''[x]/(''x''<sup>''n''</sup>&minus;1).  If ''n'' is not invertible in the base, then this scheme is not smooth.  In particular, over a field of [[characteristic (algebra)|characteristic]] ''p'', μ<sub>p</sub> is not smooth.
* The additive group '''G'''<sub>a</sub> has the affine line '''A'''<sup>1</sup> 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&nbsp;⊗&nbsp;''x''&nbsp;+&nbsp;''x''&nbsp;⊗&nbsp;1, and the inverse is given by sending ''x'' to&nbsp;&minus;''x''.
* If ''p'' = 0 in ''S'' for some prime number ''p'', then the taking of ''p''th powers induces an endomorphism of '''G'''<sub>a</sub>, and the kernel is the group scheme α<sub>p</sub>. Over an affine base such as Spec ''A'', it is the spectrum of ''A''[x]/(''x''<sup>p</sup>).
* The automorphism group of the affine line is isomorphic to the semidirect product of '''G'''<sub>a</sub> by '''G'''<sub>m</sub>, 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 '''G'''<sub>m</sub> with the automorphism group of '''G'''<sub>a</sub>.
* A smooth [[Genus (mathematics)|genus]] one curve with a marked point (i.e., an [[elliptic curve]]) has a unique group scheme structure with that point as the identity.  Unlike the previous positive-dimensional examples, elliptic curves are [[Projective variety|projective]] (in particular [[Proper morphism|proper]]).
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== Basic properties ==

Suppose that ''G'' is a group scheme of finite type over a field ''k''.  Let ''G''<sup>0</sup> be the connected component of the identity, i.e., the maximal connected subgroup scheme.  Then ''G'' is an extension of a [[étale group scheme|finite étale group scheme]] by ''G''<sup>0</sup>.  ''G'' has a unique maximal reduced subscheme ''G''<sub>red</sub>, and if ''k'' is perfect, then ''G''<sub>red</sub> 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 of a ring|spectrum]] of a commutative [[Hopf algebra]] (over a base ''S'', this is given by the relative spectrum of an ''O''<sub>S</sub>-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 variety|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 variety|abelian varieties]]. 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]] 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''<sub>''G''</sub> is a locally free [[Sheaf of modules|''O''<sub>''S''</sub>-module]] of finite rank.  The rank is a locally constant function on ''S'', and is called the order of&nbsp;''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]].

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, μ<sub>2</sub> 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''<sup>2</sup>, but over '''F'''<sub>p</sub>, it is a finite flat group scheme of order ''p''<sup>2</sup> that has either ''p'' connected components (if the curve is ordinary) or one connected component (if the curve is [[supersingular]]).  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]], where supersingular points are replaced by discs of positive radius.

==Cartier duality==

{{Main|Cartier duality}}
Cartier duality is a scheme-theoretic analogue of [[Pontryagin duality]] taking finite commutative group schemes to finite commutative group schemes.

==Dieudonné modules==

{{Main|Dieudonné module}}
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''){''F'',''V''}/(''FV''&nbsp;&minus;&nbsp;''p''), which is a quotient of the ring of noncommutative polynomials, with coefficients in [[Witt vectors]] 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 [[Sheaf of modules|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''<sub>n</sub> → ''W''<sub>n+1</sub>, 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 [[Andrew Wiles|Wiles]]'s work on the [[Shimura–Taniyama conjecture]].

== See also ==
*[[Fundamental group scheme]]
*[[Geometric invariant theory]]
*[[GIT quotient]]
*[[Groupoid scheme]]
*[[Group-scheme action]]
*[[Group-stack]]
*[[Invariant theory]]
*[[Quotient stack]]

==References==
{{Reflist}}

*{{cite book
 |editor-last=Demazure
 |editor-first=Michel
 |editor2=Alexandre Grothendieck
 |editor2-link=Alexandre Grothendieck
 | title = Séminaire de Géométrie Algébrique du Bois Marie &ndash; 1962&ndash;64 &ndash; Schémas en groupes &ndash; (SGA 3) &ndash; vol. 1 (Lecture notes in mathematics '''151''')
 | year = 1970
 | publisher = [[Springer Science+Business Media|Springer-Verlag]]
 | location = Berlin; New York
 | language = fr
 | pages = xv, 564
}}
*{{cite book
 |editor-last=Demazure
 |editor-first=Michel
 |editor2=Alexandre Grothendieck
 |editor2-link=Alexandre Grothendieck
 | title = Séminaire de Géométrie Algébrique du Bois Marie &ndash; 1962&ndash;64 &ndash; Schémas en groupes &ndash; (SGA 3) &ndash; vol. 2 (Lecture notes in mathematics '''152''')
 | year = 1970
 | publisher = [[Springer Science+Business Media|Springer-Verlag]]
 | location = Berlin; New York
 | language = fr
 | pages = ix, 654
}}
*{{cite book
 |editor-last=Demazure
 |editor-first=Michel
 |editor2=Alexandre Grothendieck
 |editor2-link=Alexandre Grothendieck
 | title = Séminaire de Géométrie Algébrique du Bois Marie &ndash; 1962&ndash;64 &ndash; Schémas en groupes &ndash; (SGA 3) &ndash; vol. 3 (Lecture notes in mathematics '''153''')
 | year = 1970
 | publisher = [[Springer Science+Business Media|Springer-Verlag]]
 | location = Berlin; New York
 | language = fr
 | pages = vii, 529
}}
*{{cite book |author1=Gabriel, Peter |author2=Demazure, Michel |title=Introduction to algebraic geometry and algebraic groups |publisher=North-Holland Pub. Co |location=Amsterdam |year=1980 |isbn=0-444-85443-6 }}
*Berthelot, Breen, Messing ''Théorie de Dieudonné Crystalline II''
*Laumon, ''Transformation de Fourier généralisée''
* {{Citation | last1=Shatz | first1=Stephen S. | editor1-last=Cornell | editor1-first=Gary | editor2-last=Silverman | editor2-first=Joseph H. | editor2-link=Joseph H. Silverman | title=Arithmetic geometry (Storrs, Conn., 1984) | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-96311-2  |mr=861972 | year=1986 | chapter=Group schemes, formal groups, and ''p''-divisible groups | pages=29–78}}
* {{Citation | last1=Serre | first1=Jean-Pierre | author1-link=Jean-Pierre Serre | title=Groupes algébriques et corps de classes | publisher=Hermann | location=Paris | series=Publications de l'Institut Mathématique de l'Université de Nancago [Publications of the Mathematical Institute of the University of Nancago], 7 | isbn=978-2-7056-1264-1 |mr=907288 | year=1984}}
*[[John Tate (mathematician)|John Tate]], ''Finite flat group schemes'', from ''Modular Forms and Fermat's Last Theorem''
* {{Citation | last1=Waterhouse | first1=William | author1-link=William C. Waterhouse | title=Introduction to affine group schemes | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=978-0-387-90421-4 | year=1979 | volume=66 | doi=10.1007/978-1-4612-6217-6 | mr=0547117}}

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[[Category:Algebraic groups]]
[[Category:Scheme theory]]
[[Category:Hopf algebras]]
[[Category:Duality theories]]