# 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 $D\left(G\right)$ of $G$ is defined by

$D\left(G\right)\left(R\right):={\mathrm{Gr}}_{R}\left(G{\otimes }_{k}R,{\mu }_{R}\right)$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 ${R}^{×}$ consisting of the invertible elements of $R$.

This definition deserves the name duality since we have

$\mathrm{hom}\left(G,D\left(H\right)\right)=\mathrm{hom}\left(H,D\left(G\right)\right)=\mathrm{hom}\left(G×H,{\mu }_{k}\right)$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. $\stackrel{^}{D}\left(G\right)$ is an étale $k$-formal group.

5. ${\mathrm{Gr}}_{k}\left(G,{\alpha }_{k}\right)=0$

6. (If $p\ne 0\right)$, ${V}_{G}$ is an epimorphism

7. (If $p\ne 0\right)$, ${V}_{G}$ 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. $D\left({G}_{\mathrm{const}}\right)$ is diagonalizable.

2. $D\left(E\right)$ is multiplicative

Revised on July 19, 2012 00:18:27 by Stephan Alexander Spahn (79.227.135.169)