A new proof of D. Popescu's theorem on smoothing of ring homomorphisms

We give a new proof of D. Popescu's theorem which says that if is a regular homomorphism of noetherian rings, then is a filtered inductive limit of smooth finite type -algebras. We strengthen Popescu's theorem in two ways. First, we show that a finite type -algebra , mapping to , has a desingularization which is smooth wherever possible (roughly speaking, above the smooth locus of ). Secondly, we give sufficient conditions for to be a filtered inductive limit of its smooth finite type -subalgebras. We also give counterexamples to the latter statement in cases when our sufficient conditions do not hold.