Popescu's theorem

In commutative algebra and algebraic geometry, Popescu's theorem, introduced by Dorin Popescu,^[1][2] states:^[3]

Let A be a Noetherian ring and B a Noetherian algebra over it. Then, the structure map A → B is a regular homomorphism 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,^[4] while an exposition of the result was provided by Richard Swan.^[5]

The usual proof of the Artin approximation theorem relies crucially on Popescu's theorem. Popescu's result was proved by an alternate method, and somewhat strengthened, by Mark Spivakovsky.^[6][7]

See also

• Ring with the approximation property