nLab Cartier duality

Contents

Idea

Let $G$ be a finite group scheme over $k$ , regarded as a sheaf of groups $G \in Sh(Ring^{op}_k)$ . Write $\mathbb{G}_m$ for the multiplicative group, similarly regarded.

Then the Cartier dual $\widehat G$ is the internal hom

\widehat G \coloneqq [G,\mathbb{G}_{m}]

of group homomorphisms, hence the sheaf which to $R \in Ring_k^{op}$ assigns the set

\widehat G \;\colon\; R \mapsto Hom_{Grp/Spec R}(G \times Spec R, \mathbb{G}_m \times Spec R)

of group homomorphisms over $Spec(R)$

This appears for instance as (Polishchuk, (10.1.11)).

Cartier duality is indeed a duality in that for any finite commutative group scheme $G$ there is an isomorphism

\widehat{\widehat{G}} \simeq G

of the double Cartier dual with the original group scheme.

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

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)

and in

Dima Arinkin, appendix in Ron Donagi, Tony Pantev, Torus fibrations, gerbes, and duality, (arXiv:math/0306213)

Discussion in the context of higher algebra (brave new algebra) is in

Last revised on September 26, 2016 at 16:51:02. See the history of this page for a list of all contributions to it.