Spahn multiplicative group scheme (Rev #2, changes)

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A Recall that the multiplicative group scheme is calledmultiplicative group schemeμ k:RR ×\mu_k:R\mapsto R^\times if assigns the to following a equivalent conditions are satisfied:kk-ring RM kR\in M_k the multiplicative group consisting of the invertible elements of RR.

  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 p¬=0)p\not =0), V GV_G is an epimorphism

  7. (If p¬=0)p\not =0), V GV_G is an isomorphism

In generalization of this group μ k\mu_k is the following notion of multiplicative group scheme:

Definition and Remmark

A group scheme is called multiplicative group scheme if the following equivalent conditions are satisfied:

  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\neq 0), V GV_G is an isomorphism

Remark

Let G constG_const dnote a constant group scheme, let EE be an étale group scheme. Then we have the following cartier duals:

  1. D(G const)D(G_const) is diagonalizable.

  2. D(E)D(E) is multiplicative

Revision on June 1, 2012 at 11:46:18 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.