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–Lück and Jaikin-Zapirain on universal localizations and universal fields of fractions of such crossed products.

在 Jaikin-Zapirain 最近的工作的基础上，我们提供了一个环是伪 Sylvester 域的同调标准，也就是说，允许所有稳定满矩阵都可逆的分数除环。我们使用该标准来研究自由理想环（冷杉）上的斜洛朗多项式环。

作为我们方法的一个应用，我们证明了具有自由无限循环和表面基团的划分环的交叉产物无条件地是伪 Sylvester 域，并且当且仅当它们允许稳定的自由取消时 Sylvester 域。这依赖于 Farrell-Jones 猜想对正常多自由基团的最近证明，并扩展了 Linnell-Lück 和 Jaikin-Zapirain 先前关于此类交叉产物分数的普遍定位和普遍领域的结果。