Because of this feature, some authors, notably Waterhouse and Milne (who followed Waterhouse),[1] develop the theory of group schemes based on the notion of group functor instead of scheme theory.
of S-schemes to the category of sets satisfying the gluing axiom; the perspective known as the functor of points.
Under this perspective, a group scheme is a contravariant functor from
For example, if Γ is a finite group, then consider the functor that sends Spec(R) to the set of locally constant functions on it.
The notion appears in particular in the discussion of a torsor (where a choice of topology is an important matter).