Affine Group Schemes

Affine group schemes are exactly the group functors constructed by solution of equations. But such a definition would be technically awkward, since quite different collections of equations can have essentially the same solutions. For this reason the official definition is postponed to the next section, where we translate the condition into something less familiar but more manageable. 

Such F are called representable, and one says that A represents F. We can now officially define an affine group scheme over k as a representable functor from k-algebras to groups. 

Theorem. Affine group schemes over k correspond to Hopf algebras over k. 

A homomorphism of affine group schemes is a natural map G -+ H for which each G(R) - H(R) is a homomorphism. We have already seen the example det GL - Gm. The Yoneda lemma shows as expected that such maps correspond to Hopf algebra homomorphisms. But since any map between groups preserving multiplication also preserves units and inverses, we need to check only that A is preserved. An algebra homomorphism between Hopf algebras which preserves A must automatically preserve S and e. 

Theorem. Characters of an affine group scheme G represented by A correspond to group-like elements in A. 

If G and H are any abelian group functors over k, we can always get another group functor Hom(G, H) by attaching to JR the group Hom(GR, Hr). This is the functorial version of Horn, and for H = Gm it is a functorial character group for finite G it is GD. In general it will not be an affine group scheme even when G and H are Cartier duality is one case where it is representable. 

Let F, G, and H be commutative affine group schemes over k. Show that homomorphisms F - Hom(G, H) correspond to natural biadditive maps F x G- H. 

Theorem. Let G be an affine group scheme represented by A. Then linear representations of G on V correspond to k-linear maps p V - V A such that... 

We call an affine group scheme G algebraic if its representing algebra is finitely generated. 

Corollary. Every affine group scheme G over a field is an inverse limit of algebraic affine group schemes. 

Theorem. Every algebraic affine group scheme over a field is isomorphic to a closed subgroup of some GL . 

Lemma. Let G be an affine group scheme over a field. Every finite-dimensional representation of G embeds in a finite sum of copies of the regular representation. 

G is an affine group scheme with G(k) = S. From (3.4) we now obtain the corresponding result here ... 

Theorem. Let G be an algebraic affine group scheme, A = k[G]. The following are equivalent ... 

Theorem. Let G be an algebraic affine group scheme. Then 7c0(Jc[G]) represents an etale group n0 G, and all maps from G to etale groups factor through the canonical map G - jr0 G. The kernel G° of this map is a connected closed normal subgroup represented by the factor ofk[G] on which s is nonzero. The construction of ic0G and G° commutes with base extension. 

Theorem. Let G be a connected affine group scheme acting as automorphisms of an algebraic group scheme T of multiplicative type. Then G acts trivially. 

Let G and H be algebraic affine group schemes. Show that every homomorphism from G to H over k is actually defined over a finite extension of k. 

Let G be a finite group scheme, H any affine group scheme. Show Hom(G, H) is representable. [Embed it in the Weil restriction of H o].]... 

The last theorem shows us how to define unipotence for arbitrary affine group schemes G is unipotent if every nonzero linear representation has a nonzero fixed vector. For this we must first define fixed vectors, but ob-... 

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