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group scheme

A group scheme is a group object in the category of schemes. Therefore it may be considered a generalization of an algebraic group. 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.

• 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)