Group scheme

2020 Mathematics Subject Classification: Primary: 14L15 Secondary: 20G35 [MSN][ZBL]

The notion of a group scheme is a generalization of the concept of an algebraic group. Let ${\rm Sch}/S$ be the category of schemes over a ground scheme $S$; a 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).

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 [DeGr], and a cohomology theory of extensions of group schemes and homogeneous spaces over group schemes has been developed [DeGa], [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 [Oo2], 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. $p$-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; 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; Finite group scheme.

Other examples of group schemes are Abelian (group) varieties [Mu].

References

[DeGa]	M. Demazure, P. Gabriel, "Groupes algébriques", 1, Masson (1970) MR0302656 MR0284446 Zbl 0223.14009 Zbl 0203.23401
[DeGr]	M. Demazure, A. Grothendieck, "Schémas en groupes. I-III", Lect. notes in math., 151–153, Springer (1970) MR0274458 MR0274459 MR0274460
[Mu]	D. Mumford, "Abelian varieties", Oxford Univ. Press (1974) Zbl 0326.14012
[Oo]	F. Oort, "Commutative group schemes", Lect. notes in math., 15, Springer (1966) MR0213365 Zbl 0216.05603
[Oo2]	F. Oort, "Finite group schemes, local moduli for abelian varieties and lifting problems" Compos. Math., 23 (1971) pp. 256–296 MR0301026 Zbl 0223.14024
[TaOo]	J. Tate, F. Oort, "Group schemes of prime order" Ann. Sc. Ecole Norm. Sup., 3 (1970) pp. 1–21 MR0265368 Zbl 0195.50801