# Group functor

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
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
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

can be described as the functor

If we take a ring, for example, , then

## 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

## References

- Waterhouse, William (1979),
*Introduction to affine group schemes*, Graduate Texts in Mathematics,**66**, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-6217-6, ISBN 978-0-387-90421-4, MR 0547117