Popescu’s theorem

In algebra, Popescu’s theorem, introduced by D. Popescu, states:[1]

Let A be a noetherian ring and B a noetherian algebra over it. Then, the structure map AB 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 AB is regular by definition and the theorem applies.

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

References

  1. Conrad & De Jong, Theorem 1.3.

External links

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