duality

# Contents

## Idea

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

## Definition

###### Definition

Let $G$ be a commutative $k$-group functor (in cases of interest this is a finite flat commutative group scheme). Then the Cartier dual $D\left(G\right)$ of $G$ is defined by

$D\left(G\right)\left(R\right)={\mathrm{Gr}}_{R}\left(G{\otimes }_{k}R,{\mu }_{R}\right)$D(G)(R)=Gr_R(G\otimes_k R,\mu_R)

Moreover we have

$\mathrm{hom}\left(G,D\left(H\right)\right)=\mathrm{hom}\left(H,D\left(G\right)\right)=\mathrm{hom}\left(G×H,{\mu }_{k}\right)$hom(G,D(H))=hom(H,D(G))=hom(G\times H,\mu_k)
Revised on August 20, 2012 14:38:18 by Urs Schreiber (89.204.138.243)