Pseudo-Sylvester domains and skew Laurent polynomials over firs

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 non-commutative field 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-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 poly-free groups and extends previous results of Linnell-Lück and Jaikin-Zapirain on universal localizations and universal fields of fractions of such crossed products. Contents 1.2. Ore and universal localizations 1.3. Weak and global dimensions 1.4. Stably freeness and stably finite rings 1.5. (Pseudo-)Sylvester domains 2. Towards Theorem A 2.1. Homological recognition principles for (pseudo-)Sylvester domains 2.2. The homological properties of D S and the proof of Theorem A 3. Applications: Theorem B and Theorem C 3.1. The Farrell-Jones conjecture and stably freeness 3.2. Examples and non-examples 3.3. Locally indicable groups and Hughes-freeness: identifying D E * G References