Cartier duality via Mittag-Leffler modules

TL;DR

This paper develops a duality theory for affine commutative group schemes with coordinate rings as flat Mittag-Leffler modules, establishing a Fourier-Mukai transform in derived categories under certain conditions.

Contribution

It constructs the Cartier duality equivalence for a broad class of group schemes using Mittag-Leffler modules and introduces a Fourier-Mukai transform in derived categories.

Findings

Cartier duality is established for affine commutative group schemes with Mittag-Leffler coordinate rings.

The dual group scheme is shown to be an ind-scheme over the base ring.

A Fourier-Mukai transform between derived categories is constructed when the base ring is Noetherian with a dualizing complex.

Abstract

We construct the Cartier duality equivalence for affine commutative group schemes $G$ whose coordinate ring is a flat Mittag-Leffler module over an arbitrary base ring $R$. The dual $G^\vee$ of $G$ turns out to be an ind-finite ind-scheme over $R$. When $R$ is Noetherian and admits a dualizing complex, we construct a Fourier-Mukai transform between quasicoherent derived categories of $G$ and of $BG^\vee$ and also between those of $G^\vee$ and $BG$.