Algebraic affine group scheme

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 . 

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. 

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. 

Corollary. Let k be algebraically closed of characteristic zero. Then all algebraic affine group schemes come from algebraic matrix groups. 

Let G be an algebraic affine group scheme. By Noether normalization (A.7) we can write k[G] as a finite module over a polynomial ring k[Xu..., X ], The n occurring here is obviously unchanged by base extension. It is uniquely determined, since it is the transcendence degree of the fraction field of k[G°]/nilpotents. Intuitively it represents the number of independent parameters involved in expressing elements of G, and we call it the dimension of G. 

Algebraic affine group scheme 24 Algebraic matrix group 29 Anisotropic torus 56 Anti-equivalence 15 Antipode 8 Arf invariant 147 Artin-Schreier theory 143 Augmentation ideal 13 Automorphism group scheme 58... 

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