# nLab Cartier duality

Contents

### Context

#### Duality

duality

Examples

In QFT and String theory

# Contents

## Idea

Cartier duality is a refinement of Pontryagin duality from 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 (Polishchuk, (10.1.11)).

###### Proposition

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.

## References

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)

and in

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.