Cartier duality is a refinement of Pontryagin duality from topological groups to group schemes.
Let be a finite group scheme over , regarded as a sheaf of groups . Write for the multiplicative group, similarly regarded.
Then the Cartier dual is the internal hom
of group homomorphisms, hence the sheaf which to assigns the set
of group homomorphisms over
This appears for instance as (Polishchuk, (10.1.11)).
Cartier duality is indeed a duality in that for any finite commutative group scheme there is an isomorphism
of the double Cartier dual with the original group scheme.
(e.g. Polishchuk, right above (10.1.11), Hida 00, theorem 1.7.1)
The classical textbook account is in chapter 1 of
- Jean Dieudonné, Introduction to the theory of formal groups, Marcel Dekker, New York 1973.
and a more recent textbook account is in section 10.1 of
or section 1.7 of
- Haruzo Hida, Geometric Modular Forms and Elliptic Curves, 2000, World scientific
lecture notes include
Generalization beyond finite group schemes is discussed in
- Amelia Álvarez Sánchez, Carlos Sancho de Salas, Pedro Sancho de Salas, Functorial Cartier duality (arXiv:0709.3735)
Discussion in the context of higher algebra (brave new algebra) is in
Revised on January 28, 2016 11:13:33
by Tim Porter