On the generalized Bykovskii presentation of Steinberg modules

TL;DR

This paper generalizes Bykovskii's presentation of Steinberg modules from integers to Gaussian and Eisenstein integers, exploring the limitations of this approach for other Euclidean number rings.

Contribution

It extends Bykovskii's presentation to specific number rings and analyzes its applicability to various Euclidean rings.

Findings

Generalization to Gaussian and Eisenstein integers achieved

The presentation does not extend to all Euclidean number rings

Provides insights into the structure of Steinberg modules over number rings

Abstract

We study presentations of the virtual dualizing modules of special linear groups of number rings, the Steinberg modules. Bykovskii gave a presentation for the Steinberg modules of the integers, and our main result is a generalization of this presentation to the Gaussian integers and the Eisenstein integers. We also show that this generalization does not give a presentation for the Steinberg modules of several Euclidean number rings.