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Cartier duality

Contents

Idea

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

Definition

Definition

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)).

Proposition

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)

References

The classical textbook account is in chapter 1 of

and a more recent textbook account is in section 10.1 of

or section 1.7 of

lecture notes include

Generalization beyond finite group schemes is discussed in

and in

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