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{{MSC|14L15|20G35}}
{{TEX|done}}


The notion of a ''group scheme'' is a generalization of the concept of an
[[Algebraic group|algebraic group]]. Let ${\rm Sch}/S$ be the category of
schemes over a ground scheme $S$; a
[[Group object|group object]] of this category is known as a group
scheme over the scheme $S$ (or a group $S$-scheme, or an $S$-scheme
group). For a group scheme $G$ over $S$ the functor of points $h_G:X\to{\rm Hom}_{\rm Sch/S}(X,G)=G(X)$ is a
contravariant functor from the category ${\rm Sch}/S$ into the category of
groups ${\rm Gr}$. The category $S-{\rm Gr}$ of group schemes over $S$ is defined as
the complete subcategory of the category of such functors formed by
the representable functors (cf.
[[Representable functor|Representable functor]]).

===Examples.===


1) An algebraic group over a field $k$ is a reduced group scheme of
finite type over $k$. (A reduced group scheme of finite type over a
field is sometimes referred to as an algebraic group.)

2) A functor which assigns to an $S$-scheme $X$ the additive (or
multiplicative) group of the ring of sections of the structure sheaf
$\Gamma(X,{\mathcal O}_X)$ is representable. The corresponding group scheme over $S$ is said
to be the additive (or multiplicative) group scheme, and is denoted by
$G_{\alpha,S}$ (or $G_{m,S}$). For any $S$-scheme $S_1$ one has 
$$G_{\alpha,S}\times_S S_1 \simeq G_{\alpha,S_1},\qquad G_{m,S}\times_S S_1 \simeq G_{m,S_1}$$
3) Each abstract
group $\Gamma$ defines a group scheme $(\Gamma)_S$, which is the direct sum of a
family of schemes $(S_g)_{g\in G}$, each one of which is isomorphic to $S$. The
corresponding functor maps an $S$-scheme $X$ to the direct sum $\Gamma^{\pi_0(X)}$,
where $\pi_0(X)$ is the set of connected components of $X$.

If $G$ is a group scheme over $S$ then, for any point $s\in S$, the fibre
$G_s = G\otimes_S k(s)$ is a group scheme over the residue field $k(s)$ of this point. In
particular, any group scheme of finite type over $S$ can be regarded
as a family of algebraic groups parametrized by the base $S$. The
terminology of the theory of schemes is extended to group schemes;
thus, one speaks of smooth, flat, finite, and singular group schemes.

For any group scheme $G$ the corresponding reduced scheme $G{\rm red}$ is also
a group scheme; the canonical closed imbedding $G{\rm red} \to G$ is a morphism of
group schemes. Each reduced group scheme of locally finite type over a
perfect field is smooth. Each reduced group scheme of locally finite
type over a field of characteristic zero is reduced (Cartier's
theorem).

Many concepts and results in the theory of algebraic groups have their
analogues for group schemes. Thus, there exists an analogue of the
structure theory of Borel–Chevalley for affine algebraic groups
{{Cite|DeGr}}, and a cohomology theory of extensions of group
schemes and homogeneous spaces over group schemes has been developed
{{Cite|DeGa}},
{{Cite|DeGr}}. On the other hand, many problems and results
specific to the theory of group schemes are connected with the
presence of nilpotent elements in the structure sheaf of both the
ground scheme and the group scheme itself. Thus, infinitesimal and
formal deformations of group schemes
{{Cite|Oo2}}, problems of lifting into zero characteristic, and
formal completion of group schemes (cf.
[[Formal group|Formal group]]) have all been studied. Group schemes
arise in a natural manner in the study of algebraic groups over a
field of positive characteristic (cf.
[[P-divisible group|$p$-divisible group]]).

The concept of an affine group scheme over an affine ground scheme $S={\rm Spec}\;(B)$
is dual to the concept of a commutative
[[Hopf algebra|Hopf algebra]]; this is the case if $G={\rm Spec}\;(A)$ is a group
scheme for which $A$ is a commutative Hopf algebra.

See also
[[Commutative group scheme|Commutative group scheme]];
[[Finite group scheme|Finite group scheme]].

Other examples of group schemes are Abelian (group) varieties {{Cite|Mu}}.

====References====
{|
|-
|valign="top"|{{Ref|DeGa}}||valign="top"| M. Demazure, P. Gabriel, "Groupes algébriques", '''1''', Masson (1970)  {{MR|0302656}} {{MR|0284446}}  {{ZBL|0223.14009}} {{ZBL|0203.23401}}    
|-
|valign="top"|{{Ref|DeGr}}||valign="top"| M. Demazure, A. Grothendieck, "Schémas en groupes. I-III", ''Lect. notes in math.'', '''151–153''', Springer (1970)     {{MR|0274458}} {{MR|0274459}} {{MR|0274460}}  
|-
|valign="top"|{{Ref|Mu}}||valign="top"| D. Mumford, "Abelian varieties", Oxford Univ. Press (1974)   {{ZBL|0326.14012}}    
|-
|valign="top"|{{Ref|Oo}}||valign="top"| F. Oort, "Commutative group schemes", ''Lect. notes in math.'', '''15''', Springer (1966)  {{MR|0213365}}  {{ZBL|0216.05603}}    
|-
|valign="top"|{{Ref|Oo2}}||valign="top"| F. Oort, "Finite group schemes, local moduli for abelian varieties and lifting problems" ''Compos. Math.'', '''23''' (1971) pp. 256–296  {{MR|0301026}}  {{ZBL|0223.14024}}    
|-
|valign="top"|{{Ref|TaOo}}||valign="top"| J. Tate, F. Oort, "Group schemes of prime order" ''Ann. Sc. Ecole Norm. Sup.'', '''3''' (1970) pp. 1–21  {{MR|0265368}}  {{ZBL|0195.50801}}    
|-
|}