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),{{Cite web|url=http://www.jmilne.org/math/CourseNotes/|title=Course Notes -- J.S. Milne}} 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 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.{{clarify|clarify topology|date=May 2018}} For example, the group scheme
:
can be described as the functor
:
If we take a ring, for example, , then
:
\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}
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).{{Cite web |url=http://people.maths.ox.ac.uk/chojecki/gdtScholze1.pdf |title=Archived copy |access-date=2018-03-26 |archive-date=2016-10-20 |archive-url=https://web.archive.org/web/20161020141125/http://people.maths.ox.ac.uk/chojecki/gdtScholze1.pdf |url-status=dead }}
See also
Notes
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References
- {{Citation | last1=Waterhouse | first1=William | author1-link=William_C._Waterhouse | title=Introduction to affine group schemes | publisher=Springer-Verlag | 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}}