Let be a commutative -group functor (in cases of interest this is a finite flat commutative group scheme). Then the Cartier dual of is defined by
where denotes the group scheme assigning to a ring its multiplicative group consisting of the invertible elements of .
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:
is diagonalizable for a field .
is the Cartier dual of an étale -group.
is an étale -formal group.
(If , is an epimorphism
(If , is an isomorphism