nLab Cartier duality

Idea

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

Definition

Definition

Let $k$ be a commutative ring.

Let $G$ be a commutative 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 in Polishchuk, (10.1.11).

Say that $G$ is finite locally free if it is an affine $k$-scheme whose coordinate ring is a finite locally free $k$-module.

Proposition

( SGA3, Exposé VIIA, §3.3) If $G$ is a finite locally free commutative $k$-group scheme, then $\widehat{G}$ is also a a finite locally free commutative $k$-group scheme, and the natural map

$$G \to \widehat{\widehat{G}}$$

is an isomorphism.

This proposition is also discussed in Polishchuk, right above (10.1.11) and Hida 00, theorem 1.7.1.

When $k$ is a field, every finite group scheme is locally free. When $G$ is no longer finite over $k$, it is still true that $G \to \widehat{\widehat{G}}$ is an isomorphism (Dieudonné 73, p. 21-22)

Related concepts

• dual abelian group scheme

• Picard scheme

• Poincaré line bundle

References

The case of finite locally free group schemes is covered in:

• Michel Demazure, Alexander Grothendieck (eds.): Schémas en groupes. Tome I: Propriétés générales des schémas en groupes, Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA3)

The classical textbook account over a field 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, World Scientific (2000)

lecture notes include

• pdf

Generalization beyond finite group schemes:

• 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

• Jacob Lurie, section 5.3 of Ambidexterity in K(n)-Local Stable Homotopy Theory
• Jacob Lurie, section 3 of Elliptic Cohomology I