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

Abstract

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