# Group scheme

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A generalization of the concept of an algebraic group. Let be the category of schemes over a ground scheme ; a group object of this category is known as a group scheme over the scheme (or a group -scheme, or an -scheme group). For a group scheme over the functor of points is a contravariant functor from the category into the category of groups . The category of group schemes over is defined as the complete subcategory of the category of such functors formed by the representable functors (cf. Representable functor).

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### Examples.

1) An algebraic group over a field is a reduced group scheme of finite type over . (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 -scheme the additive (or multiplicative) group of the ring of sections of the structure sheaf is representable. The corresponding group scheme over is said to be the additive (or multiplicative) group scheme, and is denoted by (or ). For any -scheme one has

3) Each abstract group defines a group scheme , which is the direct sum of a family of schemes , each one of which is isomorphic to . The corresponding functor maps an -scheme to the direct sum , where is the set of connected components of .

If is a group scheme over then, for any point , the fibre is a group scheme over the residue field of this point. In particular, any group scheme of finite type over can be regarded as a family of algebraic groups parametrized by the base . 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 the corresponding reduced scheme is also a group scheme; the canonical closed imbedding 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 [5], and a cohomology theory of extensions of group schemes and homogeneous spaces over group schemes has been developed [2], [5]. 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 [4], problems of lifting into zero characteristic, and formal completion of group schemes (cf. 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. -divisible group).

The concept of an affine group scheme over an affine ground scheme is dual to the concept of a commutative Hopf algebra; this is the case if is a group scheme for which is a commutative Hopf algebra.

#### References

 [1] J. Tate, F. Oort, "Group schemes of prime order" Ann. Sc. Ecole Norm. Sup. , 3 (1970) pp. 1–21 [2] M. Demazure, P. Gabriel, "Groupes algébriques" , 1 , Masson (1970) [3] F. Oort, "Commutative group schemes" , Lect. notes in math. , 15 , Springer (1966) [4] F. Oort, "Finite group schemes, local moduli for abelian varieties and lifting problems" Compos. Math. , 23 (1971) pp. 256–296 [5] M. Demazure, A. Grothendieck, "Schémas en groupes I-III" , Lect. notes in math. , 151–153 , Springer (1970)