Spahn multiplicative group scheme (Rev #2, changes)

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

$G\otimes_k k_s$ is diagonalizable.

$G\otimes_k K$ is diagonalizable for a field $K\in M_k$ .

$G$ is the Cartier dual of an étale $k$ -group.

$\hat D(G)$ is an étale $k$ -formal group.

$Gr_k(G,\alpha_k)=0$

(If $p\not =0)$ , $V_G$ is an epimorphism

(If $p\not =0)$ , $V_G$ is an isomorphism

In generalization of this group $\mu_k$ 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:

$G\otimes_k k_s$ is diagonalizable.
$G\otimes_k K$ is diagonalizable for a field $K\in M_k$ .
$G$ is the Cartier dual of an étale $k$ -group.
$\hat D(G)$ is an étale $k$ -formal group.
$Gr_k(G,\alpha_k)=0$
(If $p\neq 0)$ , $V_G$ is an epimorphism
(If $p\neq 0)$ , $V_G$ is an isomorphism

Remark

Let $G_const$ dnote a constant group scheme, let $E$ be an étale group scheme. Then we have the following cartier duals:

$D(G_const)$ is diagonalizable.
$D(E)$ 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.