Spahn multiplicative group scheme (Rev #1, changes)

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

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