In commutative algebra and algebraic geometry, Popescu's theorem, introduced by Dorin 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., a local excellent ring) and B its completion, then the map A →B is regular by definition and the theorem applies. Another proof of Popescu's theorem was given by Tetsushi Ogoma, while an exposition of the result was provided by Richard Swan.