# On rank filtrations of algebraic K-theory and Steinberg modules

## TL;DR

This paper proves a conjecture about the high connectivity of the common basis complex for fields, providing a new description using bar constructions and linking Steinberg modules to Koszul duality.

## Contribution

It confirms the high connectivity conjecture for fields and introduces a novel bar construction approach for PIDs, connecting Steinberg modules to Koszul duality.

## Key findings

- Proved the connectivity conjecture for fields.
- Provided a new bar construction description for PIDs.
- Linked Steinberg modules to Koszul duality.

## Abstract

Motivated by his work on the stable rank filtration of algebraic K-theory spectra, Rognes defined a simplicial complex called the common basis complex and conjectured that this complex is highly connected for local rings and Euclidean domains. We prove this conjecture in the case of fields. Our methods give a novel description of this common basis complex of a PID as an iterated bar construction on an equivariant monoid built out of Tits buildings. We also identify the Koszul dual of a certain equivariant ring assembled out of Steinberg modules.