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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

$$D(G)(R):={\mathrm{Gr}}_R(G{\otimes }_{k}R,{\mu }_R)$$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

$$\mathrm{hom}(G,D(H))=\mathrm{hom}(H,D(G))=\mathrm{hom}(G\times H,{\mu }_k)$$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 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. ${\mathrm{Gr}}_k(G,{\alpha }_k)=0$

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

7. (If $p\ne 0)$, $V_G$ is an isomorphism

Remark

Let $G_{\mathrm{const}}$ dnote a constant group scheme?, let $E$ be an étale group scheme?. Then we have the following cartier duals:

1. $D(G_{\mathrm{const}})$ is diagonalizable.

2. $D(E)$ is multiplicative