A Solomon-Tits theorem for rings

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.