Cartier duality is a refinement of Pontryagin duality form 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 (Polishuk, (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. Polishuk, right above (10.1.11), Hida 00, theorem 1.7.1)
A textbook account is for instance in section 10.1 of
- Alexander Polishchuk, Abelian Varieties, Theta Functions and the Fourier Transform
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 June 26, 2015 02:50:31