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 \( {\displaystyle {\mathsf {Sch}}_{S}} \) 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 \( {\displaystyle {\mathsf {Sch}}_{S}} \) 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
\( {\displaystyle SL_{2}=\operatorname {Spec} \left({\frac {\mathbb {Z} [a,b,c,d]}{(ad-bc-1)}}\right)} \)
can be described as the functor
\( {\displaystyle \operatorname {Hom} _{\textbf {CRing}}\left({\frac {\mathbb {Z} [a,b,c,d]}{(ad-bc-1)}},-\right)} \)
If we take a ring, for example, \( \mathbb {C} \) , then
\( {\displaystyle {\begin{aligned}SL_{2}(\mathbb {C} )&=\operatorname {Hom} _{\textbf {CRing}}\left({\frac {\mathbb {Z} [a,b,c,d]}{(ad-bc-1)}},\mathbb {C} \right)\\&\cong \left\{{\begin{bmatrix}a&b\\c&d\end{bmatrix}}\in M_{2}(\mathbb {C} ):ad-bc=1\right\}\end{aligned}}} \)
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
"Course Notes -- J.S. Milne".
"Archived copy" (PDF). Archived from the original (PDF) on 2016-10-20. Retrieved 2018-03-26.
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
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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