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$E_\infty$-cells and general linear groups of infinite fields

Soren Galatius, Alexander Kupers, Oscar Randal-Williams

TL;DR

The paper proves a striking vanishing line for the $E_ fty$-homology of the rank-graded general linear group algebra over connected semi-local rings with infinite residue fields, namely $H^{E_ fty}_{n,d}(old{R}_ Z)=0$ for $d<2n-2$, and analyzes the critical boundary $d=2n-2$ to obtain explicit generators tied to divided-power structures. It develops and leverages the $E_k$-cell framework, together with higher-dimensional (split) buildings, to compute $E_k$-homology and connect it to coinvariants of Steinberg modules, providing concrete results for $E_1$, $E_2$, and $E_ fty$-homology. The work yields broad applications: a unified derivation of Nesterenko–Suslin-type theorems, determinations of Milnor $K$-theory relations, and new insights into stability phenomena, pre-Bloch groups, and associated homology operations for general linear groups. These findings illuminate the unstable homology of ${ m GL}_n(A)$, relate algebraic $K$-theory filtrations to stable buildings, and offer evidence toward Rognes' conjectures about rank filtrations and their connection to $E_ fty$-structures. Overall, the paper provides a cohesive homological and operadic framework to analyze GL groups over large classes of rings, with explicit low-degree computations and structural descriptions via divided power algebras.

Abstract

We study the general linear groups of infinite fields (or more generally connected semi-local rings with infinite residue fields) from the perspective of $E_\infty$-algebras. We prove that there is a vanishing line of slope 2 for their $E_\infty$-homology, and analyse the groups on this line by determining all invariant bilinear forms on Steinberg modules. We deduce from this a number of consequences regarding the unstable homology of general linear groups, in particular answering questions of Rognes, Suslin, Mirzaii, and others.

$E_\infty$-cells and general linear groups of infinite fields

TL;DR

The paper proves a striking vanishing line for the -homology of the rank-graded general linear group algebra over connected semi-local rings with infinite residue fields, namely for , and analyzes the critical boundary to obtain explicit generators tied to divided-power structures. It develops and leverages the -cell framework, together with higher-dimensional (split) buildings, to compute -homology and connect it to coinvariants of Steinberg modules, providing concrete results for , , and -homology. The work yields broad applications: a unified derivation of Nesterenko–Suslin-type theorems, determinations of Milnor -theory relations, and new insights into stability phenomena, pre-Bloch groups, and associated homology operations for general linear groups. These findings illuminate the unstable homology of , relate algebraic -theory filtrations to stable buildings, and offer evidence toward Rognes' conjectures about rank filtrations and their connection to -structures. Overall, the paper provides a cohesive homological and operadic framework to analyze GL groups over large classes of rings, with explicit low-degree computations and structural descriptions via divided power algebras.

Abstract

We study the general linear groups of infinite fields (or more generally connected semi-local rings with infinite residue fields) from the perspective of -algebras. We prove that there is a vanishing line of slope 2 for their -homology, and analyse the groups on this line by determining all invariant bilinear forms on Steinberg modules. We deduce from this a number of consequences regarding the unstable homology of general linear groups, in particular answering questions of Rognes, Suslin, Mirzaii, and others.

Paper Structure

This paper contains 49 sections, 73 theorems, 387 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Rognes' connectivity conjecture
  3. An invariant pairing on the Steinberg module
  4. Applications to the homology of general linear groups
  5. The Steinberg module and its tensor square
  6. Pairings on Steinberg modules
  7. Anisotropy and indecomposability
  8. Overview of $E_k$-cells and $E_k$-homology
  9. Glossary and some notation from e2cellsIv3
  10. General linear groups as an $E_\infty$-algebra
  11. Higher-dimensional buildings and their split analogues
  12. The $k$-dimensional building
  13. Split buildings
  14. The Nesterenko--Suslin property
  15. Relationship with $E_k$-homology
  16. ...and 34 more sections

Key Result

Theorem A

If $A$ is a connected semi-local ring with infinite residue fields, then $H_{n,d}^{E_\infty}(\bold{R}_\mathbb{Z})=0$ for $d < 2n-2$.

Figures (2)

  • Figure 1: The $E_\infty$-homology of $\bold{R}_\mathbb{Z}$, which vanishes for $d < 2n-2$.
  • Figure 2: The $E_\infty$-homology of the pair $(\bold{N},\bold{R}_\mathbb{Z})$, which vanishes below the dotted line. On the line $d=n+1$ it just has $A^\times$ in bidegree $(1,2)$.

Theorems & Definitions (183)

  • Theorem A
  • Theorem B
  • Remark
  • Theorem C
  • Corollary D
  • Theorem E
  • Theorem F
  • Theorem G
  • Theorem H
  • Definition 2.1
  • ...and 173 more