2023/10/11 by Matthew Scalamandre, Scalamandre, Matthew
Mathematics · #Advanced Topics in Algebra #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric Topology (math.GT) #Number Theory (math.NT) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2310.07175
openalex publication_date 2023/10/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ℤ-module and as a representation. We find the rank of Steinberg when R is a finite ring, and compute the length of St2(R)⊗ℚ as a GL2(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.