Dual pairs of algebras and finite commutative group schemes

TL;DR

This paper introduces a new categorical framework for representing finite commutative group schemes using dual pairs of algebras, along with algorithms for computation and applications to Galois representations.

Contribution

It presents a novel category of dual pairs of algebras that efficiently encodes finite locally free commutative group schemes and provides computational algorithms for their manipulation.

Findings

Efficient representation of finite commutative group schemes

Algorithms for computing with dual pairs of algebras

Applications to Galois representations on finite Abelian groups

Abstract

We introduce a category of dual pairs of finite locally free algebras over a ring. This gives an efficient way to represent finite locally free commutative group schemes. We give a number of algorithms to compute with dual pairs of algebras, and we apply our results to Galois representations on finite Abelian groups.