Spahn multiplicative group scheme

Recall that the multiplicative group scheme $\mu_k:R\mapsto R^\times$ assigns to a $k$-ring $R\in M_k$ the multiplicative group consisting of the invertible elements of $R$.

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:

1. $G\otimes_k k_s$ is diagonalizable.

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

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

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

5. $Gr_k(G,\alpha_k)=0$

6. (If $p\neq 0)$, $V_G$ is an epimorphism

7. (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:

1. $D(G_const)$ is diagonalizable.

2. $D(E)$ is multiplicative