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 S c h S {\displaystyle {\mathsf {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 S c h S {\displaystyle {\mathsf {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.[clarification needed] For example, the group scheme

S L 2 = Spec ⁡ ( Z [ a , b , c , d ] ( a d − b c − 1 ) ) {\displaystyle SL_{2}=\operatorname {Spec} \left({\frac {\mathbb {Z} [a,b,c,d]}{(ad-bc-1)}}\right)}

can be described as the functor

Hom CRing ⁡ ( Z [ a , b , c , d ] ( a d − b c − 1 ) , − ) {\displaystyle \operatorname {Hom} _{\textbf {CRing}}\left({\frac {\mathbb {Z} [a,b,c,d]}{(ad-bc-1)}},-\right)}

If we take a ring, for example, C {\displaystyle \mathbb {C} } , then

S L 2 ( C ) = Hom CRing ⁡ ( Z [ a , b , c , d ] ( a d − b c − 1 ) , C ) ≅ { [ a b c d ] ∈ M 2 ( C ) : a d − b c = 1 } {\displaystyle {\begin{aligned}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{aligned}}}

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

  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

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