Resolutions for Triangular Ore Extensions

Resolutions provide important tools for studying the homology and deformation theory of non-commutative algebras, so one desires convenient, finite resolutions to work with. We construct resolutions for noncommutative PBW algebras that arise iteratively from setting quantum or skew commutators to lower order terms, like Weyl algebras, shift algebras, skew polynomial rings, many universal enveloping algebras and others. To do so, we extend a quantum symmetrization map to a chain map between resolutions, allowing us to take advantage of the twisted product resolution for twisted tensor product algebras.