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Group functor

From Wikipedia, the free encyclopedia

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

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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.[clarification needed] For example, the group scheme

can be described as the functor

If we take a ring, for example, , then

Group sheaf

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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

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Notes

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  1. ^ "Course Notes -- J.S. Milne".
  2. ^ "Archived copy" (PDF). Archived from the original (PDF) on 2016-10-20. Retrieved 2018-03-26.{{cite web}}: CS1 maint: archived copy as title (link)

References

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