nLab multiplicative group scheme

Reminder

Let $G$ be a commutative $k$-group functor (in cases of interest this is a finite flat commutative group scheme). Then the Cartier dual $D(G)$ of $G$ is defined by

$$D(G)(R):=Gr_R(G\otimes_k R,\mu_R)$$

where $\mu_k$ denotes the group scheme assigning to a ring its multiplicative group $R^\times$ consisting of the invertible elements of $R$.

This definition deserves the name duality since we have

$$hom(G,D(H))=hom(H,D(G))=hom(G\times H,\mu_k)$$

Discussion

The notion of a multipliciative group scheme generalizes

Definition and Remark

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 $G_const$ denote 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