Group scheme
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In mathematics, a group scheme is a group object (some would prefer to say just group) in the category of schemes. That is, it is a scheme G with the equivalent properties
- there is a group law expressible as a multiplication μ and inversion map ι on G; or
- G is a functor (as in the Yoneda lemma) mapping to the category of groups, rather than just sets.
There are numerous examples familiar in algebra, including the general linear group, and elliptic curves. Those are both examples of group varieties (the scheme involved is respectively an affine variety and a projective variety).
In the case of matrix multiplication, it is obvious that the multiplication formula is polynomial in the entries. To take care of matrix inversion also, one should note that as a variety the n×n invertible matrices should be considered as having
- n2 − 1
coordinates Xij and Y subject to the equation
- det(Xij)Y − 1 = 0.
Then Cramer's rule shows that matrix inversion is polynomial in the Xij and Y. That takes care of the first equivalent property; the second is essentially the well-known fact that matrices can take values in any commutative ring, and still give a ring, from which we will take the group of units.
The theory of commutative group varieties occupies the place in theory that was investigated in nineteenth century mathematics, in the search for the most general addition theorems. Every algebraic torus and abelian variety is part of the theory, as are group extensions formed from both kinds of object (which have been called quasi-abelian varieties), used in the theory of differential forms (of the second kind and third kind, in classical terminology), and the geometric forms of class field theory.
Group schemes are a source of examples of schemes that are not reduced, that is, have nilpotent 'functions' on them (other than 0). Examples can be found over finite fields: if F is a field of characteristic p, as the kernels of endomorphisms.
For example, the additive group is just the functor sending a ring R to R+, the underlying additive group of R. Over F, it is represented by
- Spec(F[X])
with μ the mapping coming from
- F[X] → F[X,Y], X → X + Y.
(See spectrum of a ring for the duality here.)
Now compute the kernel of the p-th power map, i.e.
- x → xp
as a fiber product in the category of schemes (dually, a tensor product of R-algebras.) It turns out to be
- Spec(F[t]/tpF[t]).
The class of t is nilpotent, in the underlying ring.
This phenomenon was noticed in the duality theory of abelian varieties, which in characteristic p was seen not to be something easily expressed in terms of variety theory alone. The finer aspects were first worked out by Pierre Cartier.
Reference
Jean-Pierre Serre Groupes algébriques et corps des classes.