A Solomon-Tits theorem for rings

TL;DR

This paper extends the Tits building concept to commutative rings, proving a Solomon-Tits theorem under certain conditions, and explores the Steinberg module's properties, providing new insights into algebraic and cohomological structures.

Contribution

It introduces an analog of the Tits building and Steinberg module for rings, proving a Solomon-Tits theorem and analyzing module ranks and representations.

Findings

Proved a Solomon-Tits theorem for rings satisfying specific conditions.

Computed the rank of the Steinberg module for finite rings.

Established a lower bound for the cohomology of principal congruence subgroups.

Abstract

An analog of the Tits building is defined and studied for commutative rings. We prove a Solomon-Tits theorem when $R$ either satisfies a stable range condition, or is the ring of $S$-integers of a global field. We then define an analog of the Steinberg module of $R$, and study it both as a $\mathbb{Z}$-module and as a representation. We find the rank of Steinberg when $R$ is a finite ring, and compute the length of $\text{St}_2(R)\otimes\mathbb{Q}$ as a $\text{GL}_2(R)$-representation when $R$ is uniserial. As an application of these results, we produce a lower bound for the rank of the top-dimensional cohomology of principal congruence subgroups of nonprime level.