Unruffled extensions and flatness over central subalgebras

A condition on an affine central subalgebra Z of a noetherian algebra A of finite Gelfand–Kirillov dimension, which we call here unruffledness, is shown to be equivalent in some circumstances to the flatness of A as a Z-module. Unruffledness was studied by Borho and Joseph in work on enveloping algebras of complex semisimple Lie algebras, and we discuss applications of our result to enveloping algebras, as well as beginning the study of this condition for more general algebras