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
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
The notion of a multipliciative group scheme generalizes
A group scheme is called multiplicative group scheme if the following equivalent conditions are satisfied:
$G\otimes_k k_s$ is diagonalizable.
$G\otimes_k K$ is diagonalizable for a field $K\in M_k$.
$G$ is the Cartier dual of an étale $k$-group.
$\hat D(G)$ is an étale $k$-formal group.
$Gr_k(G,\alpha_k)=0$
(If $p\neq 0)$, $V_G$ is an epimorphism
(If $p\neq 0)$, $V_G$ is an isomorphism
Let $G_const$ denote a constant group scheme?, let $E$ be an étale group scheme?. Then we have the following cartier duals:
$D(G_const)$ is diagonalizable.
$D(E)$ is multiplicative