Augmentation ideal

Let G be a finite group scheme, not necessarily commutative. Show that G is unipotent iff the augmentation ideal in the (noncommutative) algebra k[G]D is nilpotent. 

The lemma of (11.4) could be stated just in terms of the dimensions dimk(In/Jn+l), so it is true over a field k iff it is true over k. Thus it holds for the augmentation ideal (and its translates) in any smooth group. This is actually a regularity statement (11.7) much stronger than just absence of nilpotents. In particular we can construct formal Lie groups as in (11.8). 

By group theory we have M x F = F xuF and also IV x F =s G xBF under ( ,/) - (n/ /).Thus 0B a B0cBandi)0 B 4 0cB.NowCis reduced, so we already know C - B is faithfully flat Since A injects into B, we conclude that A 0C B injects into B 0c B. As fc - B is of course faithfully flat, we conclude from this and the isomorphisms that D-+E is injective. By construction D has nilpotent augmentation ideal, so D- E is actually faithfully flat. The isomorphisms then go bade to show A 0cB- B0cB faithfully flat and hence A - B faithfully flat. ... 

We first treat the extreme case where the augmentation ideal / of A is nilpotent in this case we will show B is actually free over A. Let N be the kernel of F- G, so k[N] = C = B/IB. Then xcf under... 

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... 

Natural polymers are finding increasing applications in the area of bone replacement and hard tissue augmentation. Ideally, materials used for such purpose should be biocompatible, able to mimic the three-dimensional characteristics, physical, and mechanical nature of the bone and hard tissue, able to... 

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