nLab
group scheme

Definition

A group scheme is a group object in the category of schemes.

In the functor of points formalism, a group scheme over a scheme $X$ is a functor

$$G:(\mathrm{Sch}/X{)}^{\mathrm{op}}\to \mathrm{Grp}$$G: (Sch /X)^{op} \to Grp

(where Grp is the category of discrete groups) such that the composition with the forgetful functor $F:\mathrm{Grp}\to \mathrm{Set}$ is representable.

See also formal group.

Examples

• Given any group $G$, one can form the constant group scheme $G_X$ over $X$.

• Every algebraic group is in particular a group scheme.

• The functor ${𝔾}_m$ is a group scheme given by ${𝔾}_m(S)=\Gamma (S,{𝒪}_S{)}^{\times }$. A scheme is sent to the invertible elements of its global functions.

• The functor ${𝔾}_a$ is a group scheme given by ${𝔾}_a(S)=\Gamma (S,{𝒪}_S)$ the additive group of the ring of global functions.

• ${\alpha }_p$ is the subgroup scheme of ${𝔾}_a$ of $p$-nilpotent elements.

• ${\mu }_p$ is the subgroup scheme of ${𝔾}_m$ of $p$-th roots of unity.

Morphisms

A morphism of group schemes $G\to H$ is a morphism of schemes that is a group homomorphism on any choice of values of points. This is more easily stated by saying that a morphism of group schemes must be a natural transformation between the functor of points.

Cartier dual

Suppose now that $G$ is a finite flat commutative group scheme (over $X$). The Cartier dual of $G$ is given by the functor $G^D(S)=\mathrm{Hom}(G\otimes S,{𝔾}_m\otimes S)$. The Hom is taken in the category of group schemes over $S$.

For example, ${\alpha }_p^D\simeq {\alpha }_p$.

References

• M. Artin, J. E. Bertin, M. Demazure, P. Gabriel, A. Grothendieck, M. Raynaud, J.-P. Serre, Schemas en groupes, i.e. SGA III-1, III-2, III-3

• M. Demazure, P. Gabriel, Groupes algebriques, tome 1 (later volumes never appeared), Mason and Cie, Paris 1970

• W. Waterhouse, Introduction to affine group schemes, GTM 66, Springer 1979.

• D. Mumford, Abelian varieties, 1970, 1985.

• J. C. Jantzen, Representations of algebraic groups, Acad. Press 1987 (Pure and Appl. Math. vol 131); 2nd edition AMS Math. Surveys and Monog. 107 (2003; reprinted 2007)