Cartier duality



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



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

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

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

of group homomorphisms, hence the sheaf which to RRing k opR \in Ring_k^{op} assigns the set

G^:RHom Grp/SpecR(G×SpecR,𝔾 m×SpecR) \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)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 GG there is an isomorphism

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

of the double Cartier dual with the original group scheme.

(e.g. Polishchuk, right above (10.1.11), Hida 00, theorem 1.7.1)


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 12:51:02. See the history of this page for a list of all contributions to it.