# Commutative group scheme

A group scheme $G$ over a basis scheme $S$, the value of which on any $S$-scheme is an Abelian group. Examples of commutative group schemes are Abelian schemes and algebraic tori (cf. Algebraic torus; Abelian scheme). A generalization of algebraic tori in the framework of the theory of group schemes is the following notion. A commutative group scheme is said to be a group scheme of multiplicative type if for any point $s\in S$ there is an open neighbourhood $U\ni s$ and and an absolutely-flat quasi-compact morphism $f:U_1\to U$ such that the commutative group scheme $G_1=G\times_U U_1$ is diagonalizable over $U_1$. Here, a diagonalizable group scheme is a group scheme of the form $$D_S(M) = {\rm Spec}({\mathcal O}_S(M))$$ where $M$ is an Abelian group and ${\mathcal O}_S(M)$ is its group algebra with coefficients in the structure sheaf ${\mathcal O}_S$ of the scheme $S$. In the case when $S$ is the spectrum of an algebraically closed field, this notion reduces to that of a diagonalizable group. If $M={\mathbb Z}$ is the additive group of integers, then $D_S(M)$ coincides with the multiplicative group scheme $G_{m,S}$.

Let $G$ be a group scheme over $S$ whose fibre over the point $s\in S$ is a group scheme of multiplicative type over the residue class field $k(s)$. Then there is a neighbourhood $U$ of $s$ such that $G\times_S U$ is a group scheme of multiplicative type over $U$ (Grothendieck's rigidity theorem).

The structure of commutative group schemes has been studied in the case when the basis scheme $S$ is the spectrum of a field $k$, and the commutative group scheme $G$ is of finite type over $k$. In this case the commutative group scheme contains a maximal invariant affine group subscheme, the quotient with respect to which is an Abelian variety (a structure theorem of Chevalley). Any affine commutative group scheme $G$ of such a type has a maximal invariant group subscheme $G_m$ of multiplicative type, the quotient with respect to which is a unipotent group. If the field $k$ is perfect, then $G\cong G^m\times G^n$, where $G^n$ is a maximal unipotent subgroup of $G$.

#### References

 [1] J.-P. Serre, "Groupes algébrique et corps des classes" , Hermann (1959) [2] M. Demazure, A. Grothendieck, "Schémas en groupes II" , Lect. notes in math. , 152 , Springer (1970) [3] M. Demazure, P. Gabriel, "Groupes algébriques" , 1 , Masson (1970) [4] F. Oort, "Commutative group schemes" , Lect. notes in math. , 15 , Springer (1966) [5] W. Waterhouse, "Introduction to affine group schemes" , Springer (1979)

A group scheme $G$ over a scheme $S$ is an $S$-scheme such that $G(T)$ is a group for any $S$-scheme $T$. If $G(T)$ is an Abelian, or commutative, group for all such $T$, then $G$ is called a commutative group scheme.
The multiplicative group scheme $G_{m,S}$ takes the value $\Gamma(T,{\mathcal O}_T)^*$, the group of invertible elements of the ring of functions on $T$ for each $S$-scheme $T$. The additive group scheme $G_{\alpha,S}$ takes the values $G_{\alpha,S}(t) = \Gamma(T,{\mathcal O}_T)^+$, the underlying additive group of $\Gamma(T,{\mathcal O}_T)$. A group scheme over $S$ can equivalently be defined as a group object in the category of $S$-schemes.