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Demazure, lectures on p-divisible groups, II.9, unipotent affine groups, decomposition of affine groups

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Theorem

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.

Definition and Remark
  1. A group satisfying the conditions of the previous theorem is called unipotent k-group?.

  2. Unipotent groups correspond by duality to connected formal kk-groups.

  3. The category ACu kACu_k of affine commutative unipotent groups form a thick subcategory of AC kAC_k which is stable under limits.

The following theorem is the dual to the theorem of the previos chapter.

Theorem
  1. An affine kk group is in a unique way an extension of a unipotent group by a multiplicative group.

  2. This extension splits if kk is perfect.

  3. If kk is perfect any finite group is uniquely the product of four subgroups which are respectively étale? multiplicative, étale unipotent, infinitesimal? multiplicative and infinitesimal unipotent.

  4. The category F kF_k of finite commutative kk-groups splits as a product of four subcategories: Fem kFem_k, Feu kFeu_k, Fim kFim_k, Fiu kFiu_k.

  5. The categories Feu kFeu_k and Fim kFim_k are dual to each other.

  6. The categories Fem kFem_k and Fiu kFiu_k are selfdual.

Proposition
  1. Let p=0p = 0, then Then F k=Fem kF_k=Fem_k.

  2. Let p0p\neq 0, let kk be algebraically closed. Then any commutative finite kk-group is an extension of copies of pα kp \alpha_k, pμ kp \mu_k and (/r) k(\mathbb{Z}/r\mathbb{Z})_k where rr is prime.

Corollary

If mm is a prime and GGis a finite commutative kk-group, thenm αid G=0m^\alpha id_G=0 for large α\alpha iff rk(G)rk(G) is a power of mm.

Revised on June 5, 2012 16:18:15 by Stephan Alexander Spahn (178.195.231.138)