Non-integrality of some Steinberg modules

TL;DR

This paper investigates the generation of Steinberg modules over quadratic imaginary number rings, showing non-Euclidean rings lack generation by integral apartments and linking this to the generalized Riemann hypothesis.

Contribution

It proves the non-integrality of Steinberg modules for non-Euclidean rings and establishes a criterion for generation by integral apartments under GRH.

Findings

Steinberg module not generated by integral apartments for non-Euclidean rings

Generation by integral apartments equivalent to Euclidean property under GRH

Constructs new cohomology classes in top-dimensional cohomology groups

Abstract

We prove that the Steinberg module of the special linear group of a quadratic imaginary number ring which is not Euclidean is not generated by integral apartments. Assuming the generalized Riemann hypothesis, this shows that the Steinberg module of a number ring is generated by integral apartments if and only if the ring is Euclidean. We also construct new cohomology classes in the top dimensional cohomology group of the special linear group of some quadratic imaginary number rings.