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On split Steinberg modules and Steinberg modules

Daniel Armeanu, Jeremy Miller

TL;DR

This paper establishes that the natural map from the split Steinberg module $\widetilde{St}(M)$ to the Steinberg module $St(M)$ is surjective for every finitely-generated projective module $M$ over a Dedekind domain. The authors leverage a poset framework involving summand pairs $(P,Q)$ and their subposets, together with the Church–Putman connectivity criterion and the Solomon–Tits/Charney theory, to prove surjectivity by showing high connectivity of key subposets. The main contribution extends prior rank-bounded results (e.g., for rank $\le 4$) to all ranks, providing a uniform argument that ties representation-theoretic objects to homological stability and duality theories in arithmetic groups and $K$-theory. The techniques illuminate how split and ordinary Steinberg representations interrelate via poset topology, with potential implications for duality phenomena and algebraic $K$-theory in Dedekind-domain settings.

Abstract

Answering a question of Randal-Williams, we show the natural maps from split Steinberg modules of a Dedekind domain to the associated Steinberg modules are surjective.

On split Steinberg modules and Steinberg modules

TL;DR

This paper establishes that the natural map from the split Steinberg module to the Steinberg module is surjective for every finitely-generated projective module over a Dedekind domain. The authors leverage a poset framework involving summand pairs and their subposets, together with the Church–Putman connectivity criterion and the Solomon–Tits/Charney theory, to prove surjectivity by showing high connectivity of key subposets. The main contribution extends prior rank-bounded results (e.g., for rank ) to all ranks, providing a uniform argument that ties representation-theoretic objects to homological stability and duality theories in arithmetic groups and -theory. The techniques illuminate how split and ordinary Steinberg representations interrelate via poset topology, with potential implications for duality phenomena and algebraic -theory in Dedekind-domain settings.

Abstract

Answering a question of Randal-Williams, we show the natural maps from split Steinberg modules of a Dedekind domain to the associated Steinberg modules are surjective.
Paper Structure (3 sections, 7 theorems, 15 equations)

This paper contains 3 sections, 7 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Definitions and known results
  3. Main Result

Key Result

Theorem 1.1

Let $\Lambda$ be a Dedekind domain and let $M$ be a finitely-generated projective $\Lambda$-module. The map $(P,Q) \mapsto Q$ induces a surjective homomorphism $\Tilde{St}(M) \to St(M)$.

Theorems & Definitions (10)

  • Theorem 1.1
  • Definition 2.3
  • Lemma 2.4: Church-Putman
  • Lemma 2.5
  • Theorem 2.6: Solomon-Tits
  • Theorem 2.7: Charney
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • proof