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 In mathematics, a group scheme is a type of object from Algebraic geometry, algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of Scheme (mathematics), schemes, and they generalize algebraic groups, in ...

, the notion itself involves no

scheme theory In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations ''x'' = 0 and ''x''2 = 0 define the same algebraic variety but different sc ...

. Because of this feature, some authors, notably Waterhouse and Milne (who followed Waterhouse), develop the theory of group schemes based on the notion of group functor instead of scheme theory. A

formal group In mathematics, a formal group law is (roughly speaking) a formal power series behaving as if it were the product of a Lie group. They were introduced by . The term formal group sometimes means the same as formal group law, and sometimes means one ...

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

\mathsf_S

of ''S''-schemes to the category of sets satisfying the

gluing axiom In mathematics, the gluing axiom is introduced to define what a sheaf (mathematics), sheaf \mathcal F on a topological space X must satisfy, given that it is a presheaf, which is by definition a contravariant functor ::(X) \rightarrow C to a cate ...

; the perspective known as the

functor of points In algebraic geometry, a functor represented by a scheme ''X'' is a set-valued contravariant functor on the category of schemes such that the value of the functor at each scheme ''S'' is (up to natural bijections) the set of all morphisms S \to X. T ...

. Under this perspective, a group scheme is a contravariant functor from

\mathsf_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. For example, the group scheme :

SL_2 = \operatorname\left( \frac \right)

can be described as the functor :

\operatorname_\left(\frac, -\right)

If we take a ring, for example,

\mathbb

, then :

\begin
SL_2(\mathbb) &= \operatorname_\left(\frac, \mathbb\right) \\
&\cong \left\
\end

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 In mathematics, a principal homogeneous space, or torsor, for a group ''G'' is a homogeneous space ''X'' for ''G'' in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous space for a group ''G'' is a non-e ...

(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).

Notes

References

*{{Citation , last1=Waterhouse , first1=William , author1-link=William_C._Waterhouse , title=Introduction to affine group schemes , publisher=

Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in ...

, location=Berlin, New York , series=Graduate Texts in Mathematics , isbn=978-0-387-90421-4 , year=1979 , volume=66 , doi=10.1007/978-1-4612-6217-6 , mr=0547117 Algebraic geometry

Group functor as a generalization of a group scheme

Group sheaf

See also

Notes

References