Page "Group scheme" Paragraph 20

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 < sub > S </ sub >- 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.