Smooth Group Schemes

Sheaf in fpqc topology 117 Smooth group scheme 88 Solvable group scheme 73 Spec A 41 Split torus 56 Strictly upper triangular 62 Subcomodule 23 Symplectic group 99... 

Notice the above construction is valid if G is a smooth group scheme... 

The third chapter is about finite fiat group schemes. A good understanding of finite flat group schemes will give insight in the local structure of moduli spaces of abelian varieties. Indeed, one can show that the natural map Def(A, A) — Def(Ker(A),e ) from the deformation space of a polarized abelian variety to the deformation space of the kernel of A with its canonical pairing is formally smooth (at least if the characteristic of the base field is not equal to 2). A similar... 

Thorpe-Ziegler synthesis of 3-aminoindoles with additional functional groups was used as part of the synthesis of condensed indoles [e.g., azepines (94) were obtained from 3-amino-2-benzoylindoles (93) (91JHC379) (Scheme 24)]. In these cases the nature of the substituent R is important for a smooth reaction (Ac < Bz < 2-N02-benzoyl, but no reaction when R = H). With 2-chloro-3-(/V-bromoacetyl-/V-methylamino)pyridine and o-benzoylaminobenzonitriles (95), the condensed pyridodiazepinones 97 and 99 (95H753) were obtained via intermediates 96 and 3-aminoindoles intermediate (via 98) 3-aminoindoles followed by substitution of the 2-chloro substituent by the resulting 3-amino group (Scheme 25). 

Acyl tri-isopropylsilanes containing alkene or alkyne functionality undergo smooth hydrozirconation with Schwartz reagent, in contrast to the corresponding aldehydes, which also suffer addition to the carbonyl group (Scheme 118)225. 

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. Let G be a smooth affine group scheme over a separably closed field k. Then Hl(ic/k, G) is trivial. 

Corollary. Let F - G be a quotient map of affine algebraic group schemes, and assume the kernel N is smooth. Then F(k,)- G(kt) is surjective, and there is an exact sequence... 

---