Popescu’s theorem

In algebra, Popescu’s theorem, introduced by D. Popescu, states:

Let A be a noetherian ring and B a noetherian algebra over it. Then, the structure map A →B is a regular morphism if and only if B is a direct limit of smooth A-algebras.

For example, if A is a local G-ring (e.g., local excellent ring) and B its completion, then the map A →B is regular by definition and the theorem applies.

The usual proof of the Artin approximation theorem relies crucially on Popescu's theorem.