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Demazure, lectures on p-divisible groups, II.8, multiplicative affine groups

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Remark

Let GG be a kk-group-functor. Then the following conditions are equivalent:

  1. GG is the Cartier dual of a constant group.

  2. GG is an affine kk-group and the kk-ring O(G)O(G) is generated by the morphisms Gμ kG\to \mu_k (these are called characters of GG).

Definition

A kk-group satisfying the conditions of the previous remark is called diagnalizable k-group.

Theorem

Let GG be a kk-group. Then the following conditions are equivalent:

  1. G kk sG\otimes_k k_s is diagonalizable.

  2. G kKG\otimes_k K is diagonalizable for a field KM kK\in M_k.

  3. GG is the Cartier dual of an étale kk-group.

  4. D^(G)\hat D(G) is an étale? kk-formal group.

  5. Gr k(G,α k)=0Gr_k(G,\alpha_k)=0

  6. (If p0)p\neq 0), V GV_G is an epimorphism

  7. (If p0)p\neq0), V GV_G is an isomorphism

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

  2. Multiplicative kk-groups correspond by duality to étale formal kk-groups.

  3. The category ACm kACm_k of multiplicative kk-groups forma a subcategory of the category AC kAC_k of affine commutative kk-groups which is stable under forming subgroups, quatients, extensions (the set of these properties says that the subcategory is thick) and limits.

  4. ACm kACm_k is (contravariant) equivalent to the category of Galois modules: To GG corresponds the Galois module D^(G kk s)(k s)=Gr k s(G kk s,μ k s)\hat D(G\otimes_k k_s)(k_s)=Gr_{k_s}(G\otimes_k k_s,\mu_{k_s}).

  5. If EE is an étale kk-group, then D(E)D(E) is multiplicative and D^(D(E))=E\hat D(D(E))=E. And we have D(D(E))=ED(D(E))=E. The duality is hence given by ED(E)E\to D(E) , GD(G)G\to D(G) without reference to formal groups.

Revised on July 19, 2012 00:07:30 by Stephan Alexander Spahn (79.227.135.169)