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unipotent group scheme

Idea

An element rr of a ring with multiplicative unit is called unipotent element if r1r-1 is nilpotent.

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Theorem and Definition

Let GG be an affine k-group. Then the following conditions are equivalent.

  1. The completion of the Cartier dual D^(G)\hat D(G) of GG is a connected formal group.

  2. Any multiplicative subgroup of GG is zero.

  3. For any subgroup HH of GG with H0H\neq 0 we have Gr k(H,α k)0Gr_k(H,\alpha_k)\neq 0.

  4. Any algebraic quotient of GG is an extension of subgroups of α k\alpha_k.

  5. (If p0)p\neq 0), ImV G n=e\cap Im V^n_G =e.

An affine group scheme satisfying these conditions is called unipotent group scheme.

Revised on June 13, 2012 00:23:26 by Stephan Alexander Spahn (79.227.169.45)