Group functor explained

In mathematics, a group functor is a group-valued functor on the category of commutative rings. Although it is typically viewed as a generalization of a group scheme, the notion itself involves no scheme theory. 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.

A formal group is usually defined as a particular kind of a group functor.

Group functor as a generalization of a group scheme

A scheme may be thought of as a contravariant functor from the category

Sch _{S} 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 Sch _{S} to the category of groups that is a Zariski sheaf (i.e., satisfying the gluing axiom for the Zariski topology).

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. For example, the group scheme

SL _{2} = \operatorname{Spec}\left( (Z [a,b,c,d])/((ad - bc -1 )) \right) can be described as the functor\operatorname{Hom} _{bf{CRing} }\left(\frac, -\right)If we take a ring, for example, C , then\begin{align} SL _{2(C)} & = \operatorname{Hom} _{bf{CRing} }\left(\frac, \mathbb\right) \\&\cong \left\\end

Group sheaf

It is useful to consider a group functor that respects a topology (if any) of the underlying category; namely, one that is a sheaf and a group functor that is a sheaf is called a group sheaf. The notion appears in particular in the discussion of a torsor (where a choice of topology is an important matter).

For example, a p-divisible group is an example of a fppf group sheaf (a group sheaf with respect to the fppf topology).^[2]

See also

• automorphism group functor

Notes and References

1. Web site: Course Notes -- J.S. Milne.
2. Web site: Archived copy . 2018-03-26 . 2016-10-20 . https://web.archive.org/web/20161020141125/http://people.maths.ox.ac.uk/chojecki/gdtScholze1.pdf . dead .