Resolutions for unit groups of orders

TL;DR

This paper introduces a general algorithm to construct free resolutions for unit groups of orders in semisimple rational algebras, utilizing minimal classes of quadratic forms and dual cone theory.

Contribution

It provides a novel algorithmic approach combining geometric and algebraic methods to compute resolutions of unit groups in algebraic structures.

Findings

Algorithm successfully constructs free resolutions for unit groups.

Utilizes minimal classes of quadratic forms and dual cones.

Employs Wall's perturbation lemma to derive the resolution.

Abstract

We present a general algorithm for constructing a free resolution for unit groups of orders in semisimple rational algebras. The approach is based on computing a contractible $G$-complex employing the theory of minimal classes of quadratic forms and Opgenorth's theory of dual cones. The information from the complex is then used together with Wall's perturbation lemma to obtain the resolution.