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. For example, the group scheme : can be described as the functor : If we take a ring, for example, , then :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).See also
* automorphism group functorNotes
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 Algebraic geometry