Group Scheme - Basic Properties

Basic Properties

When working over a field, one often can analyze a group scheme by treating it as an extension of group schemes with distinguished properties. Any group scheme G of finite type is an extension of the connected component of the identity (i.e., the maximal connected subgroup scheme) by a constant group scheme. If G is connected, then it has a unique maximal reduced subscheme G_red, which is a smooth group variety that is a normal subgroup of G. The quotient group G_inf is the infinitesimal quotient, and is the spectrum of a local Hopf algebra of finite rank.

Any affine group scheme is the spectrum of a commutative Hopf algebra (over a base S, this is given by the relative spectrum of an O_S-algebra). The multiplication, unit, and inverse maps of the group scheme are given by the comultiplication, counit, and antipode structures in the Hopf algebra. The unit and multiplication structures in the Hopf algebra are intrinsic to the underlying scheme. For an arbitrary group scheme G, the ring of global sections also has a commutative Hopf algebra structure, and by taking its spectrum, one obtains the maximal affine quotient group. Affine group varieties are known as linear algebraic groups, since they can be embedded as subgroups of general linear groups.

Complete connected group schemes are in some sense opposite to affine group schemes, since the completeness implies all global sections are exactly those pulled back from the base, and in particular, they have no nontrivial maps to affine schemes. Any complete group variety (variety here meaning reduced and geometrically irreducible separated scheme of finite type over a field) is automatically commutative, by an argument involving the action of conjugation on jet spaces of the identity. Complete group varieties are called abelian varieties. This generalizes to the notion of abelian scheme; a group scheme G over a base S is abelian if the structural morphism from G to S is proper and smooth with geometrically connected fibers They are automatically projective, and they have many applications, e.g., in geometric class field theory and throughout algebraic geometry. A complete group scheme over a field need not be commutative, however; for example, any finite group scheme is complete.