Connected Group Schemes

Prior to their work with the Strecker reaction the Jacobsen group reported a combinatorial approach to the discovery of coordination complexes (29). A modular approach was taken in the synthesis of libraries of potential ligands for transition metals. Four variable components were used. Two amino acids were placed at positions 1 and 2, with a turn element connecting these groups (Scheme 9). The... 

The chain-growth condensation polymerization leading to aromatic polyether can be applied to the synthesis of a well-defined poly(ether sul-fone) by the condensation polymerization of 25, which is different from other monomers for chain-growth condensation polymerization in that the nucleophilic site and electrophilic site are on each benzene ring connected with an electron-withdrawing group, a sulfonyl group (Scheme 94). In the polymerization of 25 in the presence of an initiator and 18-crown-6 in sulfolane at... 

The last result might suggest that our introduction of n0 was actually unnecessary for studying group schemes. But in fact, though it could be avoided in the connected case, it is exactly what we need to analyze the general case. 

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. 

If char(k) = 0, all finite group schemes are in fact etale (11.4), and the other types do not occur. When char(lc) = p, however, we know examples of all four types Z/qZ with q prime to p is etale with etale dual i, while Z/pZ is etale with connected dual pp and vice versa, and p = a is connected with connected dual. The Galois theory of (6.4) describes the first two types, and also (after dualizing) the third. The fourth requires a theory of its own the groups are classified by modules over a certain ring, Dieudonne modules. ... 

Separable algebras, besides describing connected components, are related to a familiar kind of matrix and can lead us to another class of group schemes. One calls an n x n matrix g separable if the subalgebra k[p] of End(/c") is separable. We have of course k[g] k[X]/p(X) where p(X) is the minimal polynomial of g. Separability then holds iff k[g] k = /qg] a fc(Y]/p(.Y) is separable over k. This means that p has no repeated roots over k, which is the familiar criterion for g to be diagonalizable over (We will extend this result in the next section.) Then p is separable in the usual Galois theory sense, its roots are in k, and g is diagonalizable over k,. 

Theorem. An abelian matrix group H consists of separable matrices iff the group scheme G corresponding to H is of multiplicative type. IfH is connected, G is a torus... 

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. 

Other abelian group schemes of course can have connected groups of... 

Corollary. If char(k) = 0, then every unipotent algebraic group scheme is connected. 

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. 

One technical point should be mentioned. Let G be the group scheme determined by N, and Gs and Gu the subgroups determined by N, and Nu. It would a priori be possible for G, and G to have nontrivial (finite connected) intersection even when N3 n Nu = e. By (8.3), however, that does not in fact happen here. Thus G is itself the direct product of G, and Gu. 

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. 

Other abelian group schemes of course can have connected groups of automorphisms on G , for instance, G acts by xt— ax. 

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