Pseudo-Sylvester domains and skew Laurent polynomials over firs

TL;DR

This paper establishes a homological criterion for pseudo-Sylvester domains and applies it to skew Laurent polynomial rings over free ideal rings, demonstrating their properties in specific algebraic constructions.

Contribution

It introduces a homological criterion for pseudo-Sylvester domains and applies it to analyze skew Laurent polynomial rings over free ideal rings, extending previous results.

Findings

Crossed products of division rings with certain groups are pseudo-Sylvester domains.

Such crossed products are Sylvester domains if they have stably free cancellation.

The results rely on recent proofs of the Farrell--Jones conjecture for specific groups.

Abstract

Building on recent work of Jaikin-Zapirain, we provide a homological criterion for a ring to be a pseudo-Sylvester domain, that is, to admit a division ring of fractions over which all stably full matrices become invertible. We use the criterion to study skew Laurent polynomial rings over free ideal rings (firs). As an application of our methods, we prove that crossed products of division rings with free-by-{infinite cyclic} and surface groups are pseudo-Sylvester domains unconditionally and Sylvester domains if and only if they admit stably free cancellation. This relies on the recent proof of the Farrell--Jones conjecture for normally poly-free groups and extends previous results of Linnell--L\"uck and Jaikin-Zapirain on universal localizations and universal fields of fractions of such crossed products.