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A Solomon-Tits Theorem for Rings

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posted on 2024-05-09, 16:53 authored by Matthew Joseph Scalamandre
An analogue 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 analogue 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 St(2,R)?Q as a 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.

History

Date Created

2024-04-15

Date Modified

2024-05-09

Defense Date

2024-03-04

CIP Code

  • 27.0101

Research Director(s)

Andrew Putman

Committee Members

Mark Behrens|Christopher Schommer-Pries|Nicholas Salter

Degree

  • Doctor of Philosophy

Degree Level

  • Doctoral Dissertation

Language

  • English

Library Record

006584563

OCLC Number

1433094609

Publisher

University of Notre Dame

Program Name

  • Mathematics

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