duality

# Contents

## Idea

Cartier duality is a refinement of Pontryagin duality form topological groups to group schemes.

## Definition

###### Definition

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 (Polishuk, (10.1.11)).

###### Proposition

Cartier duality is indeed a duality in that for any $G$ as above there is an isomorphism

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

of the double Cartier dual with the original group scheme.

## References

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)

and in

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

Revised on May 14, 2014 04:10:09 by Urs Schreiber (77.251.114.72)