Connected affine group scheme

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. 

Connected affine group scheme 51 Connected component of G 51 Connected set, connected component 157 Constant group scheme 16, 45 Continuous function 157 Continuous -action 48 Coseparable coalgebra 53 Crossed homomorphism 137... 

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. 

This is actually our second decomposition theorem for abelian groups in (6.8) we decomposed finite abelian group schemes into connected and etale factors. Moreover, that result is of the same type, since by (8.5) we see it is equivalent to a decomposition of the dual into unipotent and multiplicative parts. As this suggests, the theorem in fact holds for all abelian affine group schemes. To introduce the version of duality needed for this extension, we first prove separately a result of some interest in itself. 

Theorem. Let G be an affine algebraic group scheme over a field. Assume G is smooth and connected, and let H be a proper closed subgroup. Then dim H < dim G. 

Theorem. (Lang) Let k be a finite field, and G an affine algebraic group scheme which is connected. Then Hl(k/k, G) is trivial. 

Theorem. Let k be a field of characteristic zero. Let G be a connected affine algebraic group scheme acting linearly on V. A subspace W of V is stable under G iff it is stable under Lie(G). 

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