Notes on locally free class groups

TL;DR

This paper discusses Swan's theorem relating locally free class groups of maximal orders in central simple algebras to ideal class groups, and extends it to separable algebras, aiding understanding of integral group representations.

Contribution

It explains Swan's theorem and generalizes it to separable algebras, making the concepts accessible to non-specialists and relevant for integral representations of finite groups.

Findings

Swan's theorem links class groups of maximal orders to ideal class groups.

The generalization applies to separable algebras, broadening its applicability.

The article serves as an accessible introduction to the topic.

Abstract

A theorem of Swan states that the locally free class group of a maximal order in a central simple algebra is isomorphic to a restricted ideal class group of the center. In this article we discuss this theorem and its generalization to separable algebras for which it is more applicable to integral representations of finite groups. This is an expository article with aim to introduce the topic for non-specialists.