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 [math]\displaystyle{ \mathsf{Sch}_S }[/math] 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 [math]\displaystyle{ \mathsf{Sch}_S }[/math] 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
- [math]\displaystyle{ SL_2 = \operatorname{Spec}\left( \frac{\mathbb{Z}[a,b,c,d]}{(ad - bc - 1)} \right) }[/math]
can be described as the functor
- [math]\displaystyle{ \operatorname{Hom}_{\textbf{CRing}}\left(\frac{\mathbb{Z}[a,b,c,d]}{(ad - bc - 1)}, -\right) }[/math]
If we take a ring, for example, [math]\displaystyle{ \mathbb{C} }[/math], then
- [math]\displaystyle{ \begin{align} SL_2(\mathbb{C}) &= \operatorname{Hom}_{\textbf{CRing}}\left(\frac{\mathbb{Z}[a,b,c,d]}{(ad - bc - 1)}, \mathbb{C}\right) \\ &\cong \left\{ \begin{bmatrix}a & b \\ c & d \end{bmatrix} \in M_2(\mathbb{C}) : ad-bc = 1 \right\} \end{align} }[/math]
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
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
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